Numerical Properties of Pseudo-effective Divisors
by
Brian Lehmann
B.S., Yale University, June 2004
Certificate of Advanced Study, Cambridge University, June 2005
Submitted to the Department of Mathematics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
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Author ...............................
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2
Numerical Properties of Pseudo-effective Divisors
by
Brian Lehmann
Submitted to the Department of Mathematics
on April 23, 2010, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
Suppose that X is a smooth variety and L is an effective divisor. One of the main
goals of birational geometry is to understand the asymptotic behavior of the linear
series ImL| as m increases. The two most important features of the asymptotic
behavior - the litaka dimension and the litaka fibration - are subtle and difficult
to work with. In this thesis we will construct approximations to these objects that
depend only on the numerical class of L. The main interest in such results arises
from the Abundance Conjecture which predicts that the Iitaka fibration for Kx is
determined by its numerical properties.
In the second chapter we study a numerical approximation to the Iitaka dimension
of L. For a nef divisor L, this quantity is a classical invariant known as the numerical
dimension. There have been several proposed extensions of the numerical dimension
to pseudo-effective divisors in [Nak04] and [BDPP04]. We show that these proposed
definitions coincide and agree with many other natural notions. Just as in the nef
case, the numerical dimension v(L) of a pseudo-effective divisor L should measure
the maximum dimension of a subvariety W C X such that the "positive restriction"
of L is big along W.
In the third chapter, we analyze how the properties of the Iitaka fibration OL for
L are related to the numerical properties of L. Although the numerical dimension
detects the existence of "virtual sections", it does not have a direct relationship with
the Iitaka fibration. However, we do construct a rational map that only depends on the
numerical class of L and approximates the Jitaka fibration. This rational map is the
maximal possible fibration for which a general fiber F satisfies v(LIF) = 0. Thus, this
chapter recovers and extends the work of [Eck05] from an algebraic viewpoint. Finally,
we use the pseudo-effective reduction map to study the Abundance Conjecture. 1
Thesis Supervisor: James McKernan
Title: Professor of Mathematics
'This material is based upon work supported under a National Science Foundation Graduate
Research Fellowship.
4
Acknowledgments
First of all, I would like to thank the many people who have contributed to my
mathematical development. My interest in the subject was first sparked by T. Rona,
L. Gallagher, and J. Blumenthal, whose courses challenged and fascinated me. Since
then, many mathematicians have spent their time and energy to mentor me. In
particular, I would like to thank I. Frenkel, J. Starr, and K. Kedlaya. I have also
benefited from the REU experience under J. Morrow and A. Bialostocki. I would like
to show my appreciation to M. Behrens for administering my qualifying exams and
S. Kleiman and C. Xu for sitting on my thesis committee. Finally, a very heartfelt
thanks to my two advisors: I. Coskun, for generously giving his time to teach me
the foundations of algebraic geometry, and J. McKernan, for his kind mentorship and
support through my early research years and his extensive help with this thesis.
Personally, I would like to thank my family and friends for their support. My
parents and sisters have been very encouraging. Liz Robertson kindly corrected many
grammatical mistakes in this paper. Special thanks go to my wife, who has tirelessly
supported my Ph.D. efforts, and my son, who has not cared a whit about my work.
Most of all, I would like to thank my savior Jesus. Without him, none of what I do
would be worthwhile.
6
Contents
1 Introduction
2
Comparing Numerical Dimensions
........................
2.1 Background ....
2.1.1 R-divisors . . . . . . . . . . . . . . . . . . . ..
2.1.2 Asymptotic Valuations and Multiplier Ideals .
2.1.3 B ase Loci . . . . . . . . . . . . . . . . . . . ..
2.1.4 V-Birational Models . . . . . . . . . . . . . ..
2.1.5 M ovable Curves . . . . . . . . . . . . . . . . ..
2.2 Divisorial Zariski Decomposition . . . . . . . . . . . ..
2.2.1 Movable Cone of Divisors . . . . . . . . . . ..
2.2.2 Birational Properties . . . . . . . . . . . . . ..
2.3 Restricted Volume . . . . . . . . . . . . . . . . . . ..
2.4 The Restricted Positive Product.............
2.4.1 Approximating Big Divisors by Ample Divisors
2.4.2 Definition and Basic Properties.... . . . . .
2.4.3 Alternate Definitions . . . . . . . . . . . . . ..
2.5 Twisted Linear Series . . . . . . . . . . . . . . . . . ..
2.5.1 Restricted Twisted Linear Series . . . . . . . ..
2.5.2 Comparisons with Other Asymptotic Invariants
. . . . .... . . . . . . .
2.6 Nakayama Constants....
2.6.1 Restricted Nakayama Constants . . . . . . . ..
2.7 The Restricted Numerical Dimension . . . . . . . . ..
3 Pseudo-effective Reduction Maps
3.1 Background . . . . . . . . . . . . . . .
3.1.1 Birational Geometry . . . . . .
3.1.2 Relations and Quotients . . . .
3.2 Previous Work . . . . . . . . . . . . .
3.2.1 Nef Reduction . . . . . . . . . .
3.2.2 Quotienting by Movable Curves
3.2.3 Analytic Approach . . . . . . .
3.3 Generic Quotients . . . . . . . . . . . .
3.4 L-trivial Reduction Map
3.5 The Restricted Numerical Dimension for Curves .............
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3.6
3.7
Pseudo-effective Reduction Map . . . . . . . . . . . . . .
3.6.1 Properties of the Pseudo-effective Reduction Map
3.6.2 Abundant Divisors . . . . . . . . . . . . . . . . .
Relationship to the Minimal Model Program . . . . . . .
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Chapter 1
Introduction
Suppose that X is a smooth variety and L is an effective divisor. One of the main goals
of birational geometry is to understand the asymptotic behavior of the linear series
|mLJ as m increases. The two most important features of the asymptotic behavior the Iitaka dimension and the Iitaka fibration - are subtle and difficult to work with. In
this thesis we will construct approximations to these objects that depend only on the
numerical class of L. These approximations provide a foundation for understanding
the discrepancy between asymptotic behavior and predictions of numerical invariants.
The main interest in such results arises from the Abundance Conjecture, one of the
key components of the birational classification of varieties. The conjecture states that
the Iitaka fibration of the canonical divisor Kx is precisely determined by numerical
properties. We will show later on that the general framework laid out here provides
additional insights in the special case of the canonical divisor.
In the second chapter we study a numerical approximation to the Jitaka dimension
of L known as the numerical dimension v(L). For nef divisors L, this is a classical
notion that has been extensively studied. [Nak04] and [BDPPO4] have proposed
several extensions of the numerical dimension to pseudo-effective divisors. The main
result of this chapter is that these proposed definitions coincide and agree with many
other natural notions.
Theorem 2.0.2. Let X be a normal variety and let L be a pseudo-effective divisor.
In the following, W will denote an intersection of very general very ample divisors,
and A some fixed sufficiently ample divisor. Then the following quantities coincide:
Volume conditions:
1. max
k E Z>o limsupmwo ho(X,tmLJ+A) > 0
2. max{dim W7limo volxIw (L + EA) > 0}.
3. max{dim
WI infp y-x voljF(P,($*L)I)
> 0} where $ varies over all birational
maps such that no exceptional center contains W and WV denotes the strict
transform of W.
Positive product conditions:
4. max{k E Z;>oI(Lk) $ 0}.
5. max{dimW|(L)xw is big }.
Geometric condition:
6. min
dimW
$*L - EE is not pseudo-effective for any e > 0
where # : Blw(X) -+ X and Ox(-E) = - 1Iw - OBx J
By convention, if L is big we interpret this expression as returningdim(X).
This common quantity is known as the numerical dimension of L, and is denoted
v(L). It only depends on the numerical class of L.
The restricted volume volxIw is explained in Definition 2.3.3, and the restricted
positive product (-)xiw is described in Section 2.4.
Just as in the nef case, the numerical dimension of a pseudo-effective divisor
measures the maximum dimension of a subvariety W C X such that the "restriction"
of L is big along W. An important subtlety is that "restriction" no longer means Liw.
Since L is no longer nef, the positivity of L along W is best measured by throwing
away the contributions of the base locus of L.
The numerical dimension satisfies several important properties:
Theorem 2.0.4. Let X be a normal variety, L a pseudo--effective divisor.
1. We have 0 < v(L) < dim(X) and r(X, L)
v(L).
2. v(L) = dim(X) iff L is big, and v(L) = 0 iff P,(L) = 0.
3. When L is nef, v(L) coincides with the usual numerical dimension.
4. If L' is pseudo-effective, then v(L + L') > v(L).
5. If f : Y -+ X is any surjective morphism from a normal variety Y then
v(f*L) = v(L).
6. We have v(L)
v(P,(L)).
7. Suppose that f : X -+ Z has connected fibers, and F is a very general fiber of
f. Then u(L) v(L|IF) + dim(Z).
We also define a restricted numerical dimension vxlv(L) which measures the positivity of L along a subvariety V. The restricted numerical dimension satisfies analogues of Theorems 2.0.2 and 2.0.4.
Definition 2.0.6. Let X be a normal variety, V a subvariety not contained in
Sing(X), and L a pseudo-effective divisor such that V g B_(L). We define the
restricted numerical dimension vxIv(L) to be
vxiv(L) := max{dim Wlimvolxlw(L + cA) > 0}
where W is an intersection of V with very general very ample divisors and A is
some fixed ample divisor. The restricted numerical dimension only depends on the
numerical class of L.
In the third chapter, we construct a fibration that approximates the Iitaka fibration
of L but only depends on the numerical class of L. This fibration is known as the
pseudo-effective reduction map and was previously defined by [Eck05].
Theorem 3.0.8 ([Eck05], Proposition 1.5 and Definition 4.1). Let X be a smooth
projective variety and L a pseudo-effective R-divisor on X. There is a birational
model $ : Y -* X and a morphism r : Y - Z with connected fibers satisfying
1. For a general fiber F of 7r, we have v(LIF) = 0.
2. The pair (Y, ir) is the maximal quotient satisfying (1): if #' : Y' -+ X is a
birationalmap and 7' : Y' -+ Z' a morphism with connected fibers that satisfies
(1), then there is a dominant rational map @: Z' -- + Z such that r $ o ' as
rational maps.
The pair (Y, -F) is determined up to birational equivalence and depends only on the
numerical class of L. We call it the pseudo-effective reduction map associated to L.
Since [Eck05] obtains this result from an analytic perspective, the main contribution of this paper is to recover and extend the work of [Eck05] using algebraic
techniques. We will discuss the relationship with Eckl's work in Section 3.2.3. Our
approach yields a more explicit description of the pseudo-effective reduction map.
Theorem 3.0.9. Let X be a smooth projective variety and L a pseudo-effective Rdivisor on X. The pseudo-effective reduction map for L is the generic quotient of X
by all movable curves C satisfying vx1c(L) = 0.
The pseudo-effective reduction map has the following important properties:
Theorem 3.0.10. Let X be a smooth variety and let L be a pseudo-effective divisor.
1. Suppose that 4 : Y -+ X is a birational map. The pseudo-effective reduction
map for #*L is birationally equivalent to the pseudo-effective reduction map for
L.
2. If K(X, L) > 0 then the Iitaka fibration for L factors birationally through the
pseudo-effective reduction map.
3. The pseudo-effective reduction map for P,(L) is birationally equivalent to the
pseudo-effective reduction map for L.
4. If ,(X, L) > 0, there is a birational model $ :1W -+ X and a morphism f
W -+ Z birationally equivalent to the pseudo-effective reduction map such that
there is a divisor D on Z and an effective divisor E on W with p*L ~Q f*D+E
and the section rings R(X, L) = R(Z, D) coincide.
A pseudo-effective divisor L is said to be abundant if ,(X, L) = v(L). It is wellknown that abundant nef divisors have many special geometric properties. It turns
out that these properties generalize to pseudo-effective divisors in a natural way. In
particular, L is abundant iff the pseudo-effective reduction map for L is birationally
equivalent to the Iitaka fibration (as shown in [Eck05]). The importance of abundance
derives from the following reformulation of the Abundance Conjecture:
Conjecture 3.7.3. Let (X, A) be a kit pair. Then Kx + A is abundant.
The pseudo-effective reduction map naturally leads to an inductive approach for
the Abundance Conjecture. Using the work of [Amb04), we show the following result:
Theorem 3.0.11. Let (X, A) be a kit pair. Assume that Kx + A is pseudo-effective,
and let f : X -- + Z denote the pseudo-effective reduction map. If Conjecture 3.7.3
holds on Z, then Kx + A is abundant.
In this approach to the Abundance Conjecture, the key question is whether the
pseudo-effective reduction map for Kx +A maps to a variety of smaller dimension. By
Theorem 3.0.9, this question can be answered by finding curves satisfying a numerical
condition. Similar work has appeared in the recent preprint [Siu09].
Chapter 2
Comparing Numerical Dimensions
Suppose that X is a smooth variety and L an effective divisor. One of the main
goals of birational geometry is to understand the asymptotic behavior of the linear
series ImL Ias m increases. When L is a big divisor, many features of the asymptotic
behavior are captured by invariants that can be calculated numerically. However, for
an arbitrary pseudo-effective divisor L the numerical invariants no longer predict the
behavior of sections.
In this chapter we define the numerical dimension v(L) of L. The numerical
dimension is an approximation to the Iitaka dimension of L that only depends on the
numerical class of L. In a loose sense it measures the discrepancy between predictions
of numerical invariants and the actual asymptotic behavior of sections. For a nef
divisor L, the numerical dimension is a classical invariant.
Definition 2.0.1. Let X be a normal projective variety of dimension n, L a nef
divisor. Fix an ample divisor A. The numerical dimension of L is defined to be
v(L) := max{k E Z;>o/ILk - An-
0}.
Thus, the numerical dimension of a nef divisor L is the maximal dimension of
a subvariety W of X such that Liw is big. [Nak04] and [BDPPO4] propose several
extensions of the numerical dimension to pseudo-effective divisors. Our main result
is that the proposed definitions of numerical dimension coincide and agree with many
other natural notions. (Since some of the definitions in the following theorem are
quite involved, we will recall them later in the paper.)
Theorem 2.0.2. Let X be a normal variety and let L be a pseudo-effective divisor.
In the following, W will denote an intersection of very general very ample divisors,
and A some fixed sufficiently ample divisor. Then the following quantities coincide:
Volume conditions:
1. max {k E Z>o limsupm,
ho(X, LmLJ +A) > 0}.
2. max{dim W limo volxiw(L + eA) > 0}.
3. max{dim Wjinfh.yx volFjV(P,($*L)|W) > 0} where $ varies over all birational
maps such that no exceptional center contains W and W denotes the strict
transform of W.
Positive product conditions:
4. max{k E Z->oI(L)
0}.
5. max{dimW(L)xlw is big }.
Geometric condition:
6. mn {dimW
$ *L - eE is not pseudo-effective for any e > 0
where $: Blw(X) -> X and Ox(-E)
I w - OBwx
J
By convention, if L is big we interpret this expression as returning dim(X).
This common quantity is known as the numerical dimension of L, and is denoted
v(L). It only depends on the numerical class of L.
The restricted volume volxlw is explained in Definition 2.3.3, and the restricted
positive product (-)xlw is described in Section 2.4.
Remark 2.0.3. The numerical dimension also admits a natural interpretation with
respect to separation of jets, reduced volumes, and the other invariants considered in
[ELM+09].
Just as in the nef case, the numerical dimension of a pseudo-effective divisor
measures the maximum dimension of a subvariety W c X such that the "restriction"
of L is big along W. An important subtlety is that "restriction" no longer means Liw.
Since L is no longer nef, the positivity of L along W is best measured by throwing
away contributions of the base locus of L.
There are two ways of making this intuition precise. One way to measure positivity is to use intersection products. The positive product of [BDPPO4] gives
a precise method of calculating intersections while discounting the contributions
of the base locus of L. Another way of measuring positivity is to calculate the
rate of growth of sections. Since the base locus of L may increase the dimension of H 0 (W, mLlw), it is better to work with the image of the restriction map
HO(X, Ox(mL)) -> H 0 (W, Ow(mLIw)). These subspaces more accurately reflect
the positivity of L along W.
The numerical dimension satisfies several important properties:
Theorem 2.0.4. Let X be a normal variety, L a pseudo-effective divisor.
1. We have 0 < v(L) < dim(X) and rs(X, L)
2. v((L) = dim(X) iff L is big, and v(L)
=
v(L).
0 iff P,(L) = 0.
3. When L is nef, v(L) coincides with the usual numerical dimension.
4. If L'
is pseudo-effective, then v(L + L') > v(L).
5. If f : Y --+ X is any surjective morphism from a normal variety Y then
v(f*L) = v(L).
6. We have v(L) =v(P,(L)).
7. Suppose that f X -+ Z has connected fibers, and F is a very general fiber of
f. Then v(L) < v(L|IF) + dim(Z).
Remark 2.0.5. Since bigness is an open condition, one might expect that the nu1
merical dimension is lower semicontinuous as a function on NE (X). This turns out
to be false; see Example 2.7.6.
As we study different numerical invariants, we will come across two common
themes. First, we will define numerical invariants for pseudo-effective divisors by
taking the limit of asymptotic invariants of nearby big divisors. Since asymptotic
invariants of big divisors can be calculated numerically, this approach ensures that
our invariant is numerical in nature. It also allows us to avoid the difficulties of
working with pseudo-effective divisors of negative Iitaka dimension. In some sense
the numerical invariants will measure the existence of "virtual sections" of L.
The second theme is that it is useful to study not only numerical invariants on
X but also restricted numerical invariants along subvarieties V. In particular, we
can define the notion of a restricted numerical dimension of L along a subvariety V.
Just as in the non-restricted case, the restricted numerical dimension should measure
the maximal dimension of a very general subvariety W C V such that the "positive
restriction" of L is big along W.
Definition 2.0.6. Let X be a normal variety, V a subvariety not contained in
Sing(X), and L a pseudo-effective divisor such that V tB (L). We define the
restricted numerical dimension vxiv(L) to be
vxIv(L) := max{dim W limvolxiw(L + eA) > 0}
where W is an intersection of V with very general very ample divisors and A is
some fixed ample divisor. The restricted numerical dimension only depends on the
numerical class of L.
The restricted numerical dimension satisfies (slightly weaker) analogues of Theorems 2.0.2 and 2.0.4. It does not occur in the classical analysis because when L
is a nef divisor we have valv(L) = vv(Liv). Nevertheless, it is useful to have the
additional flexibility for a general pseudo-effective divisor. The restricted numerical
dimension satisfies several compatibility relations:
Proposition 2.0.7. Let X be a normal variety, V a normal subvariety not contained
in Sing(X), and L a pseudo-effective divisor such that V 9 B_L). Then
1. vxlv( L) < v(L).
2. vxjv(L) < v(Ljv).
3. If H is a general very ample divisor and vxiv(L) < dim V, then vXlvnH
vxIv(L).
2.1
Background
All schemes will lie over the base field C. A variety will always be an irreducible
reduced projective scheme. We will usually restrict our attention to normal varieties
X.
Throughout the paper we will be working with R-Cartier divisors. Since the
behavior of R-divisors can be subtle, a couple of remarks are in order. In the literature,
often asymptotic invariants (such as the volume) are only defined for Q-divisors. It
is then checked that these invariants can be extended to continuous functions on the
entire big cone. It is easy to see that this continuous extension coincides with the
naive extension of the definition to big R-divisors. Thus, there is usually no difficulty
in considering asymptotic invariants of big divisors.
The behavior of R-divisors on the pseudo-effective boundary is much more subtle.
However, as mentioned in the introduction, we will define numerical invariants by
taking a limit of invariants of nearby big divisors. Thus, we will avoid the potential
difficulties of the boundary case. We refer to [Nak04) as a reference for properties of
pseudo-effective R-divisors.
2.1.1
R-divisors
Suppose that L = E aiLi is an R-Cartier divisor (henceforth "divisor"). As usual,
the support of L, denoted Supp(L), is defined to be the set of points x for which
a local equation of one of the Li is not a unit in OXx. We define the round-down
[LJ
Ei [aij Li. Then [L] is an integral Cartier divisor, and we denote the difference
{L} : L - [LJ. We will use the notation b( [L]) to denote the base ideal of the linear
system [L] Iand Bs([L]) to denote the set-theoretic base locus.
We say that two R-divisors L, L' are linearly equivalent if L - L' is a principal
divisor. That is, we must have {L} = {L'} and [L] - [L'] = div(f) for some
meromorphic function f on X. We will write L
L' to denote linear equivalence.
Then the complete linear system associated to L is
|L| =|[LjI+{L}.
Suppose that |LI is not empty. Then sections of I[L]I define a rational map <D|L|
X -- + P(HO(X, Ox([LJ))). We denote the closure of the image by WL. The Iitaka
dimension r,(X, L) is defined to be
K(X, L) = max{dim WmL
unless
ImLI
ImLI $
0}
is empty for every m, in which case we set i,(X, L) = -oo.
We can also
construct an Iitaka fibration 7r : X -- + Z for L: it is characterized up to birational
equivalence by the property that dim(Z) = i(X, L) and K(F, L|F) = 0 for a very
general fiber F of 7r.
We will say that two R-divisors L, L' are R-linearly equivalent (respectively Qlinearly equivalent) if L - L' is a R-linear combination (resp. Q-linear combination) of
principal divisors. We write this relation as L ' L' (resp. L ~,Q L'). We write L -- L'
to denote numerical equivalence, and use the notation [L] to denote the class of L in
N1 (X). AP(X) will denote the space of rational equivalence classes of codimension p
cycles and NP(X) will denote the space of numerical classes of codimension p cycles.
The R-stable base locus BR(L) is defined to be
BR(L) :=
n{ Supp(C) IC > 0
and C ~R L}.
This is always a Zariski-closed subset of X; we do not associate any scheme structure
to it. When L is a Q-divisor, this is the same as the usual stable base locus, but
for a general R-divisor it will be smaller than the corresponding intersection over all
Q-linearly equivalent divisors (see [BCHM10], Lemma 3.5.3).
2.1.2
Asymptotic Valuations and Multiplier Ideals
Asymptotic valuations of big R-divisors are an important example of a numerical
invariant. Throughout this section we will let v denote a discrete valuation of the
function field K(X) of X. When such a valuation measures the the order of vanishing
along a subvariety Z of X, we will denote it by vz. For a Cartier divisor D, we define
v(D) to be v(f) where f is any local defining equation for D on a open set intersecting
the center of v. We then extend the valuation by R-linearity to arbitrary R-divisors.
Similarly, for an ideal sheaf I C Ox, we can define the valuation v(I) by first passing
to a principalization of I and then using the definition for divisors. On an open affine
U c X intersecting the center of v, this is equivalent to taking the valuation of an
element f E 1(U) "general" in the sense that it is a general linear combination of the
generators of I(U).
Definition 2.1.1. Let X be a normal variety. Suppose that L is a divisor with
,(L) > 0 and v is a discrete valuation on X. We define the asymptotic order of
vanishing of L along v as follows:
1
rn
v(IILII) = min -v(D).
mEZ>o
DEImLi
We will almost always restrict our attention to the case when L is big. In this
case, asymptotic valuations only depend on the numerical class of L.
Theorem 2.1.2 ([ELM+09), Theorem A and Lemma 3.3). Let X be a normal variety.
If L is big, then
v(||L||=
minv(D).
D=L
D>0
Furthermore, v(|| - ||) is continuous on the big cone.
Asymptotic valuations share a close relationship with the asymptotic multiplier
ideal. Multiplier ideals are most commonly used for vanishing theorems. Thus, for
the sake of clarity we will only work with multiplier ideals when the underlying variety
X is smooth. We begin by recalling the definition of a multiplier ideal:
Definition 2.1.3. Suppose that X is smooth and that L is an effective R-divisor.
Let # : Y -* X be a log resolution of the pair (X, L). We define the multiplier ideal
of L to be the ideal sheaf
j(X, L) :=
y (Ky/x
- LOp L.
Since #,0Oy(Ky/x) = Ox, this sheaf is in fact an ideal sheaf. It turns out to be
independent of the choice of resolution. In some sense the multiplier ideal sheaf
measures the singularities of L.
The asymptotic multiplier ideal sheaf associated to L measures the singularities
of a general divisor D with D - L. We first recall the definition for Q-divisors:
Definition 2.1.4. Let X be smooth. Suppose that L is a Q-divisor with K(X, L) > 0.
Consider the set of ideals {J(X, D)} where D ~R L (or equivalently D -( L). It
is shown in [Laz04] that this is a directed set under the inclusion relation. Since
the underlying rings are Noetherian, there is a unique maximal element. We define
J(X, jLj|) to be this maximal element.
Remark 2.1.5. Suppose that L is a big Q-divisor. It turns out that the asymptotic
multiplier ideal sheaf J(X, lILI|) is the unique ideal sheaf satisfying the valuation
conditions
vz(J(|L||)) = Lvz(||L||) - codim(Z) + I
where Z ranges over all subvarieties in X. In particular, the asymptotic multiplier
ideal sheaf only depends on the numerical class of L.
We would like to extend the notion of asymptotic multiplier ideal to R-divisors.
When L is big, we can define the asymptotic multiplier ideal sheaf by perturbing by
an ample divisor A.
Definition 2.1.6. Let X be smooth. Suppose that L is a big R-divisor. Consider
the set of ideal sheaves {J(IIL - All)} as A varies over all ample divisors such that
L - A is a big Q-divisor. This forms a directed set under the inclusion relation, so
as before this set has a unique maximal element. We define J(IL|) to be this ideal
sheaf.
Remark 2.1.7. There is some effective divisor D ~R L such that J(IL||) = J(D).
To see this, first choose an ample divisor A sufficiently small so that J(IL|) =
J(|IL - A||) and L - A is a Q-divisor. We may then take D = D' + A' where
D' ~R L - A and A' ~R A are sufficiently general.
Note also that J(||Ll) is the unique ideal satisfying the valuation criteria
vz(J(||Ll)) = vz(||LHl) - codim(Z) + I I
where Z ranges over all subvarieties in X. Therefore this sheaf is an invariant of the
numerical class of L.
The following theorem gives a basic comparison between asymptotic multiplier
ideals and base loci. It is the analogue for R-divisors of [Laz04],Theorem 11.2.21.
Theorem 2.1.8. Let X be smooth and let L be a big R-divisor on X. Fix a very
ample Z-divisor H on X such that H + LDJ is ample for every divisor D supported
on Supp(L) with coefficients in the set [-3, 3]. Suppose that b is a sufficiently large
positive integer so that [bLj -(Kx+(n+1)H) is numerically equivalent to an effective
Z-divisor G. Then for every n > b we have
J(lmLII)
® Oy(-G)
c b( [mL)|).
Proof. The condition on H guarantees that for m > b we can write
[mLj - G
[mLj -
LbLj + Kx + (n + 1)H
((mn-b)L+ A)+Kx+nH
for some ample R-divisor A. By applying Nadel vanishing and Castelnuovo-Mumford
regularity, we find that
Ox(LmLj) 09 (Ox(-G) 0 J(I(m - b)L|))
is globally generated for m > b. Since J(lmLI) c J(11(m - b)LI|), this proves the
theorem.
2.1.3
Base Loci
We next analyze some variants of the R-stable base locus BR(L) of an effective divisor
L. The stable base locus can exhibit quite subtle behavior. For example, it is not an
invariant of the numerical class of L. The situation improves greatly if we perturb
by an ample divisor. This approach to invariants was first considered in [NakOO], and
studied more systematically in [ELM+05).
Definition 2.1.9. The augmented base locus is
B+(L) :=
BR(L - A).
A ample
Note that B+(L) D BR(L). [ELM+05] verifies that the augmented base locus is
equal to BR(L - A) for a sufficiently small ample divisor A. Thus B+(L) is a Zariskiclosed subset of X and it only depends on the numerical class of L. Note that we do
not associate any scheme structure to the augmented stable base locus.
Example 2.1.10. An R-divisor L is big iff B+(L) is not equal to X.
For the second variant, we add on a small ample divisor.
Definition 2.1.11. The restricted base locus is
U
B_(L)=
BR(L+A).
A ample
It is clear that B_ (L) c Ba(L) and that the restricted base locus only depends on
the numerical class of L. We do not associate any scheme structure to the restricted
base locus.
Example 2.1.12. An R-divisor L is nef iff B_(L) is empty.
Unlike the augmented base locus, the restricted base locus is probably not a
Zariski-closed subset (although no examples are known of such pathological behavior).
However, it is a countable union of subvarieties due to the following theorem.
Theorem 2.1.13 ([Nak04],V.1.3). Let X be a smooth variety and L be a pseudoeffective divisor. There is an ample Z-divisor A such that
U Bs(FmL
+ A)
B_ (L).
The restricted base locus is closely related to asymptotic valuations.
Theorem 2.1.14 ([ELM+09),Theorem B). Suppose that X is a smooth variety and
L is a big divisor. Let v be a discrete valuation of the function field of X with center
Z on X. Then v(||L||) > 0 iff Z C B-(L).
We immediately obtain two important corollaries.
Corollary 2.1.15. Suppose that X is smooth and L is a big R-divisor. Then as sets
U V(J(i1mLJl)) = B_(L).
Corollary 2.1.16. Suppose that # Y --+ X is a birational map of smooth projective
varieties and L is a pseudo-effective R-divisor on X. Then we have an equality of
sets
B_(#*L) = #<B_(L).
We will need a condition that describes when a subvariety V sits in "general
position" with respect to the perturbed base loci of a divisor L.
Definition 2.1.17. Suppose that V C X is a subvariety. We define the V-pseudoeffective cone Psefv(X) to be the subcone of NE (X) generated by classes of divisors
L with V g B-(L). We define the V-big cone Bigy(X) to be the cone generated by
classes of divisors L with V g B+(L).
One may verify that Psefv(X) is closed and Bigy(X) is its interior. Note also
that Llv is pseudo-effective whenever [L) E Psefv(X). Although the following interpretation of V-bigness is just a rephrasing of the definitions, the different perspective
will sometimes be useful.
Definition 2.1.18. Suppose that V C X is a subvariety. If L is an effective divisor
such that Supp(L) -6 V we say L >v 0.
The relationship with the earlier criteria is given by a trivial lemma.
Lemma 2.1.19. Suppose that V C X is a subvariety. If L is a V-big divisor, then
L ~R L' for some L' >v 0.
2.1.4
V-Birational Models
Suppose that X is a normal variety and V is a subvariety. In order to study the
geometry of a divisor L along V with respect to a birational morphism # : Y -+ X,
we need to be careful about how V intersects the exceptional centers of #. We will
restrict ourselves to the following situation:
Definition 2.1.20. Let X be a normal variety and V a subvariety not contained in
Sing(X). Suppose that # : X -> X is a birational map from a normal variety X such
that V is not contained in any #-exceptional center. Let V denote the strict transform
of V. We say that (X, V/) or # : X -> X is a V-birational model for (X, V). When
both X and V are smooth, we say that (X, V) is a smooth V-birational model.
The V-big cone satisfies natural compatibility statements for V-birational models.
Proposition 2.1.21. Let X be a normal variety and V a subvariety not contained
in Sing(X). Suppose that # : X -> X is a V-birational model. If L is a V-big divisor
then #*L is a V-big divisor. Similarly, if L is a V-big divisor then #,L is V-big.
Proof. We start with the second statement. Let H be an ample divisor on X. Then
L -e#*H is still V-big for sufficiently small e > 0. By Lemma 2.1.19, there is a divisor
L - e#*H such that D >i 0. Since V is not contained in any #-exceptional
D
center, it is also true that #,D >v 0. But then L O#D + eH is V-big.
D + eH where H is ample and
To show the first statement, as before write L
D 2v 0. Since #*D is V-pseudo-effective, it suffices to check that #*H is V-big.
Lemma 2.1.22 below shows that B+(A) is contained in the union of the #-exceptional
locus and Sing(X). By assumption this set does not contain V.
Lemma 2.1.22. Let $ : Y -+ X be a birational map of normal varieties. If A is
an ample divisor on X, then B+(#*A) is contained in the union of the #-exceptional
locus and Sing(Y).
Proof. Let @: Y -+ Y be a desingularization of Y that is an isomorphism away from
Sing(Y). Then certainly V)-B+(#A) c B+($/*#*A). We can apply Theorem 2.3.7
to Y to see that B+($*#*A) is precisely the (# o @)-exceptional locus. Since this set
l
is the preimage of the #-exceptional locus and Sing(Y), the lemma follows.
2.1.5
Movable Curves
Definition 2.1.23. Let X be a projective variety. The cone of nef curves NM 1 (X)
is the subcone of N 1 (X) that is dual to NEI(X).
Definition 2.1.24. Let X be a variety and C be an irreducible reduced curve on X.
If C is a member of a flat algebraic family of curves dominating X, I will say that C
is a movable curve.
Movable curves play a key role in birational geometry. The following theorem
gives a concrete link between movable curves and positivity notions for divisors.
Theorem 2.1.25 ([BDPPO4], Theorem 1.5). Let X be a projective variety. The
cone of nef curves NM1 (X) is the closure of the cone generated by classes of movable
curves.
2.2
Divisorial Zariski Decomposition
For a pseudo-effective divisor L on a surface, there is a classical decomposition of
L into a "positive part" and a "negative part" due to Zariski and Fujita. There
have been many attempts to generalize this decomposition in higher dimensions.
One important option was developed independently by Nakayama ([Nak04]) and by
Boucksom ([Bou04]). In this section we outline the basics of this theory and then
prove a couple important facts about the decomposition.
Both references consider only smooth varieties X. Unless noted otherwise, the
proofs cited below will work for normal varieties X with no changes.
Definition 2.2.1. Let X be a normal variety and L be a pseudo-effective divisor.
For any prime divisor F, we define
r (L) = limvr(||L +
40
cAll).
One may verify that this definition is independent of the choice of A.
Remark 2.2.2. Since vr(| - ||) is continuous on the big cone, when L is big
o-r( L) =_vr,(||L||).
Thus or is a natural extension of the notion of asymptotic order of vanishing to the
entire pseudo-effective cone. Although or is continuous on the big cone, in general it
is only upper-continuous along the pseudo-effective boundary.
The following basic result about or is the key to defining the divisorial Zariski
decomposition.
Theorem 2.2.3 ([Nak04],[Bou04]). Let X be a normal variety. For any pseudoeffective L, there are only finitely many prime divisors F with or(L) > 0.
This allows us to make the following definition:
Definition 2.2.4. Let X be a normal variety, L a pseudo-effective divisor. We define:
N, (L)= E
P,(L) = L - N,(L)
ar(L)1'
The decomposition L = N,(L) + P,(L) is called the divisorial Zariski decomposition
of L.
Remark 2.2.5. We will soon demonstrate that Supp(N,(L)) is exactly the divisorial
components of B-(L). Since the restricted base locus represents the obstruction to L
being nef, we may think of the positive part P,(L) as being "nef in codimension 1."
In this sense, it is a codimension 1 analogue of a Zariski decomposition for surfaces.
Example 2.2.6. When L is nef, L = P,(L).
divisor of a blow-up then E = N,(E).
In contrast, if E is an exceptional
Example 2.2.7. If X is a smooth surface, then the divisorial Zariski decomposition
of L coincides with the usual Zariski decomposition. In particular, the support of
N, (L) is a union of curves with negative self-intersection matrix.
The following proposition records the basic properties of the divisorial Zariski
decomposition.
Proposition 2.2.8 ([Nak04]). Let X be a normal variety, L a pseudo-effective divisor.
1. The negative part N,(L) depends only on the numerical class of L.
2. N,(L)
0 and r(X, N,(L)) = 0.
3. Supp(N,(L)) is precisely the divisorialpart of B_{ L).
4. H 0 (X, Ox(LmP,(L)])) -> H 0 (X, Ox(LmLj)) is an isomorphism for every m >
0.
Proof. All four properties are proved for smooth varieties X in [Nak04). Since vanishing theorems are used in some of the proofs, we will check the normal case explicitly.
Property (1) is a consequence of Theorem 2.1.2, which shows that asymptotic valuations are numerical invariants on the big cone. To check Property (3), let # : Y --+ X
be a desingularization of X. Fix an ample A on X; we want to analyze
#--1 B_ (L)
=
UB,
#*L + -#*A
Let T denote the #-exceptional locus. For any point y of Y not contained in T,
Lemma 2.1.22 shows that there is an expression #*A ~ H + E where H is ample,
1
E is effective, and y is not contained in the support of E. Thus, # B_(L) and
B_ (*L) coincide away from T. In particular, the two sets share the same divisorial
components outside of T, finishing the proof of Property (3). Properties (2) and (4)
l
then follow immediately from the smooth case.
2.2.1
Movable Cone of Divisors
Note that N, (L) = 0 iff B_ (L) has no divisorial components. This simple observation
leads to a different perspective on the divisorial Zariski decomposition.
Definition 2.2.9. Let X be a normal variety. The movable cone Mov 1 (X) C N 1 (X)
is the closure of the cone generated by the classes of all effective divisors L such that
B (L) has no divisorial components.
The following proposition helps clarify the terminology:
Proposition 2.2.10 ([Bou04], Proposition 2.3). Given any element a in the interior
of Mov (X), there is a birationalmap # : Y -* X and an ample divisor A on Y such
that [#,A] = a.
The positive part P,(L) of the divisorial Zariski decomposition can be understood
as a "projection" of L onto the movable cone.
Proposition 2.2.11 ([Nak04], Proposition 111.1.14). Let X be normal, L pseudoeffective. If D is an effective divisor such that L - D is movable, then N,(L)
D.
We will also need a version that accounts for a subvariety V.
Proposition 2.2.12. Let X be normal, V a subvariety, and L a V-pseudo-effective
divisor. If M is a movable divisor then L >v M iff P,(L) v M. If furthermore
L - M is V-big, then P,(L) - M is also V-big.
Proof. First suppose that P,(L) v M. Since L is V-pseudo-effective, no component
of N,(L) contains V. Thus L v ll. Conversely, suppose L M + E with E v 0.
Since M is movable, N,(L) < E. Thus E - N,(L) is still effective and does not
contain V in its support, showing that P,(L) >v M. The final statement follows
from the first by applying Lemma 2.1.19 to M - A for a small ample divisor A. E
2.2.2
Birational Properties
Suppose that # : Y -> X is a birational map. If some #-exceptional divisors are
centered in B (L), then B-(#*L) will have additional codimension 1 components.
Thus, we obtain the following birational relation:
Proposition 2.2.13 ([Nak04], 111.5.16). Let X be a normal variety and L a Vpseudo-effective divisor. Suppose # : Y -> X is a birational map. Then N,(#*L) #*N,(L) is effective and $-exceptional.
The divisorial Zariski decomposition should really be considered as an object on
all birational models of X simultaneously (in other words, as a b-divisor). We expect
the decomposition on higher models to satisfy particularly nice properties. Sometimes
we are in a very close analogue to the surface case.
Definition 2.2.14. Let X be a normal variety and L be a pseudo-effective divisor.
We say that L has a Zariski decomposition if there is a birational map
#
: Y -> X
from a smooth variety Y such that P,(#*L) is nef. We say that L has a rational
Zariski decomposition if N,(#*L) and P,(#*L) are Q-divisors.
An example due to Nakayama ([Nak04], Section IV.2) shows that Zariski decompositions do not always exist even for big divisors. Nevertheless, there is a sense in
which the positive part P,(#*L) becomes "more nef" as we pass to higher models
# : Y -+ X. We make this intuition explicit in the following proposition. In fact, we
will make the comparison in the context of a subvariety V.
Proposition 2.2.15. Let X be smooth, V a subvariety, and L a V-big divisor with
L >v 0. There is a fixed divisor G so that for any sufficiently large m there is a
Xm -+ X centered in B+(L) and a big and nef divisor Nm
V-birational model #
on X, with
1
Nm <- P,(#* L) :p N + -#* G
where Vm denotes the strict transform of V under #m.
The proof is based on the comparison between multiplier ideals and base loci in
Theorem 2.1.8. The condition L >v 0 is a technical requirement due to the fact that
we work with Q-linear equivalence. We will often apply this proposition in the context
of R-linear equivalence so that Lemma 2.1.19 makes this condition redundant.
Proof. Fix a very ample Z-divisor H and an integer b as in Theorem 2.1.8. Thus, for
any m > b we have
j(11mLI|) OOx(-G) C b(I[mL]J).
Recall that G can be chosen to be any effective Z-divisor numerically equivalent to
[bLj - (Kx + (n + 1)H). In particular, by choosing b large enough, we may choose
G so that the base locus of IG| is contained in B+(L). Since this set does not contain
V, we may ensure that G >v 0.
Let #, : Xm -+ X be a resolution of the ideals b(I LmL j) and J(I|mL1). Note that
each #m is centered in B+ (L) and is thus V-birational. We write #-1 b(I
JLj
I) Oym=
Oym(-Em) and #MJ(||mLl) - Orm = Oym(-Fm). We also define the big and nef
divisor Mm:= m*L - Em -#*{mL}.
We know that Fm + #*G > Em for all sufficiently large m. Replacing G by G + L
allows us to take into account the fractional part of mL so that
Fm + $* G > Em + #* {mrTL}
for large m. Note that we still have that G >v 0. Furthermore, since L is Vbig we know that Fm >m 0. Thus, the inequality in the equation above is a Vminequality. Furthermore N,(m#*L) 2 m Fm. In all, we get P,(m*L)
+
#* G. Dividing by m and setting Nm : Mm/nm, we get P,(#*L) ~ Nm + # *G.
The inequality Nm, < P,(q*mL) follows from Proposition 2.2.12 and the fact that
D
Em + #*{mL}> 2m 0.
2.3
Restricted Volume
Perhaps the most basic numerical invariant that describes the asymptotic behavior
of sections is the volume vol(L) of a big divisor. In this section we will explore
a variant known as the restricted volume. This notion originated in the work of
Hacon-McKernan and Takayama. [ELM+09] systematically develops the theory of
this invariant, and we will often refer to it.
Definition 2.3.1. Suppose that X is a projective variety, V is a subvariety of X,
and F is a coherent sheaf. We define the space H0 (X IV,F) to be the image of the
restriction map of sections of F from X to V. That is,
H (X|V,F) := Im(H 0 (X, F)
H (V,Fv)).
Example 2.3.2. Let X be a variety and V be a subvariety of X. Suppose that F is
a locally free sheaf and L a Z-divisor satisfying L >v 0. We will show that the map
F -+ F(L) induces an injection
H0 (X|IV,F) -* H (X|V F(L))
Since Iv is prime, the only associated prime to Oy is Iv itself. By assumption V is
not contained in Supp(L). Thus the map Ov -- Ov(D) is injective. Tensoring by F
is exact, so the map F~v -> F(D)Iv is also injective. Since the commuting diagram
HC(XF)
-+
H 0 (X,F(D)
H 0 (V,Flv)
-+
H 0 (V,F(D)|v)
has injective rows, any element in the kernel of HO(X, F) -+ HO(XIV, F(D)) must
also lie in the kernel of the restriction map to V.
We now focus on the case when F = Ox(LLJ) for an R-divisor L. Just as the
volume measures the rate of growth of the space of sections of L, the restricted volume
measures the rate of growth of space of restricted sections of L to V.
Definition 2.3.3. Suppose that X is a projective variety, V is a d-dimensional subvariety of X, and L is a divisor. We define the restricted volume volxiv(L) to be
volxIv (L)
lim sup h0(XLIV Ox(LrnLJ))
m-.oo
md/d!
Example 2.3.4. When L is an ample Z-divisor, the restricted volume can be calculated by taking intersections. Since H 1 (X, I(V)®0x(mL)) vanishes form sufficiently
large, we find volxiv(L) = volv(Llv) = L' - V.
The restricted volume gives us a more precise tool for understanding the numerical
properties of a divisor L. As with the other quantities we consider, it turns out to be
a numerical and birational invariant.
Proposition 2.3.5 ([ELM+09],Lemma 2.4). Let X be a normal variety, V a subvariety not contained in Sing(X), and L any divisor on X. Suppose that $ : (X,V ) -+
=volxIv(L).
(X, V) is a V-birational model. Then vol jf(4*L)
1
Theorem 2.3.6 ([ELM+09), Theorem A). Let X be a normal variety, V a subvariety
not contained in Sing(X), and L a V-big divisor. Then volxiv(L) > 0, and this
quantity only depends on the numerical class of L. Furthermore, volx1v(L) varies
continuously on the V-big cone.
In some cases, the limit as L approaches the boundary of the V-big cone can
also be understood. More precisely, the restricted volume identifies the maximal
components of B+(L).
Theorem 2.3.7 ([ELM+09], Theorem C). If X is smooth and L is a divisor on
X, then B+(L) is the union of all positive dimensional subvarieties V such that
volxIv(L) = 0.
In general, it does not seem possible to say much for V C B+(L), even when V
is not contained in the stable base locus of L. [ELM+091 gives an example of a base
point free L and a curve C such that volxic is not continuous near L.
3
Example 2.3.8 ([ELM+09], Example 5.10). Let f : X -+ P be the blow-up along a
line 1. The exceptional divisor E is isomorphic to PI x P, and we identify flE with
projection onto the first component. We let L be the pullback of a hyperplane H on
Ps. Note that L is big and base point free but B+(L) = E.
0
Let C be a curve of type (2,1) in E. By Theorem 2.3.5 h (X C, Ox(mL)) =
0
m+ 1, so that the restricted volume is 1. However, consider the
h (P3l, Op3(mH))
L
- 1 E. Since the restricted volume for an ample divisor is just
ample divisors Lj:=
an intersection number, volxic(Lj) = Lj - C = 2 + . Thus the restricted volume is
not continuous near L.
Despite this difficulty [P T09] gives the following characterization of when volxIv (L) >
0 for a general V:
Theorem 2.3.9 ([PT09], Theorem 1.1). Let X be a smooth variety. Suppose L is a
Q-divisor with r(X, L) > 0 and suppose V passes through a general point of X. Then
volxlv(L) > 0 iff the Iitaka fibration of L is generically finite on V.
2.4
The Restricted Positive Product
Fujita realized that one can study the asymptotic behavior of sections of a big divisor
L by analyzing the ample divisors sitting beneath L on higher birational models.
The positive product (developed in [Bou04] and [BDPPO4]) is a construction that
encapsulates this approach to asymptotic behavior. In this section, we'll consider the
restricted positive product constructed in [BFJ09].
. .L2. Lk)xlv is a
Conceptually speaking, the restricted positive product (L1 V. In contrast
along
Li
the
of
intersection"
"positive
the
represents
that
class on V
to the usual intersection product Li - L2 - ... -Lk - V, the restricted positive product
throws away the contributions of the base loci of the Li. The result is a rational
equivalence class of cycles that gives a more precise measure of the positivity of the
Li along V. In order to avoid working with the singularities of V, we will define the
restricted positive product as a collection of classes on smooth V-birational models
that are compatible under pushforward.
2.4.1
Approximating Big Divisors by Ample Divisors
Any big divisor can be written L ~ A+E for an ample divisor A and effective divisor
E. In general, the ample part A will not have a particularly close relationship to L.
However, if one allows birational modifications, it turns out that one can find such an
expression so that A is a close approximation to L. We first make our terminology
precise.
Definition 2.4.1. Let X be a projective variety and L a big divisor. A Fujita
approximation of L is a birational model # : Y -+ X and a decomposition #*L ~
A + E where A is an ample divisor and E is an effective divisor. We will sometimes
refer to A as the Fujita approximation if the other data are implicit.
The basic realization of Fujita is that the volume of a big divisor L can be approximated arbitrarily well by the volume of a Fujita approximation A. In fact, many
other asymptotic constructions can also be calculated using Fujita approximations.
In this section, our goal is to show that for smooth varieties the positive product can
be computed using only Fujita approximations
#
Y -+ X centered in B_ (L). The
first step is the following proposition:
Proposition 2.4.2. Let X be a smooth variety, V a subvariety of X, and L a V-big
divisor. Then there is a sequence of V-birational maps
#,
: X,
-+ X centered in
B_(L) and Fujita approximations #*L ~- Am + En with E_ 2f 0 such that the
limit limm-+O #mA exists and satisfies
lim
m,*Am
~ P, L).
This proposition is well-known if we allow centers in B+(L) (for example, it is a
consequence of Proposition 2.2.15). An easy rescaling argument allows us to pass to
B_ (L).
Proof. Recall the result of Proposition 2.2.15. Assuming that L >v 0, the proposition
allows us to find #m centered in B+(L) and big and nef divisors Nm such that Nm <5f
#* L. Furthermore, [m,*Nm] converges to [P,(L)] in the natural metric on N(X).
As a consequence, we see that limm-o #mNm = Po,(L).
In order to construct the Am, we perturb the situation slightly. Since L is V-big,
we may write L ~ H + F for some ample divisor H >v 0 and some V-big divisor
F with F v 0. Define D : H + c- i F.B ) Since D =+1 L + j+o' H, there is a
containment B+(Dj) c B-(L).
Since Dj v 0, we may apply Proposition 2.2.15 to Dj to find a sequence of V#*eDj
birational maps #m, centered in B_(L) and big and nef divisors Nm1 <such that
= P,(Dj).
lim #m,j*Nmj
m-ooc
#mjDj
F. Since F is V-big, we may remove a small
ample and add it to Nnj to write #*L ~R Amj + Emj where
Consider
#*,jL
~R
+ -#*
" Armi is ample and limn-oo
* Emj
#mnj* Arnj
= P,(Dj).
is Vmj-big and so by Lemma 2.1.19 we may assume Em,
Note that P,(Dj) converges to P,(H +F) as
a subsequence of the Amj will suffice.
2.4.2
j
Vm2
0.
increases. Since P,(H +F) ~R P, (L),
Definition and Basic Properties
The restricted positive product (L 1
.
L 2 - ... - Lk)xIv is a rational equivalence class
of cycles on V that represents the intersections of the "positive parts" of the Li with
V. Since we do not make any assumptions on the singularities of V, it is best to
think of the restricted positive product as a collection of classes associated to smooth
V-birational models (X, V) that is compatible under pushforward.
Since we will work with cycles of arbitrary codimension, we will need the following
notation:
Definition 2.4.3. Let X be a smooth variety of dimension n. Suppose that K, K'
are two classes in An-k(X). We write K >- K' if K - K' is contained in the closure of
the cone generated by effective cycles of dimension k. We will use the same notation
for numerical equivalence classes of cycles.
The following lemma from [BFJ09] lies at the heart of the restricted positive
product.
Lemma 2.4.4. Let X be a smooth variety, V a smooth subvariety, and L 1 ,..., Lk
V-big divisors. Consider the classes
#
(N1 -N
2
....Nk
)E Ak(V)
# : (X,1) -+ (X, V) varies over all smooth V-birational models, the Ni are
nef, and E := #*Li - Ni is a Q-divisor satisfying E, 2 0. Then these classes form
where
a directed set under the relation -<.
Corollary 2.4.5. The classes
m (N1 - N2.
admit a unique maximum as
a oVe
Ak ( V
(,Vf) varies over all smooth V-birational models.
Proof. Following [BFJ09], we first prove that the numerical classes
[# (N 1 - N 2
....
.
Nk -V )]
N'(V)
vary in a compact set. Suppose that we choose ample divisors Ai on X such that
Ai- Li is ample and effective. Then for any Ni as in the statement we have Ni #*A.
By the push-pull formula every class satisfies
# (N1 - N2
-....-
Nk -V
) -- A1...
Ak -V
The set of all numerical classes satisfying 0 - [a] -- [A1 ... Ak - V] is compact since
Nk(V) is finite-dimensional.
Next, we note that the set of rational equivalence classes K satisfying 0 - K is a
proper cone. In particular, suppose that a, is a directed set of rational equivalence
classes under -3whose numerical classes converge. Then the maximum of the ai under
is uniquely defined as a rational equivalence class.
E
-
The restricted positive product is defined as the maximum class occurring in the
previous corollary.
Definition 2.4.6. Let X be a smooth variety and V be a smooth subvariety. Let
L 1 , L 2 ,. . . , Lk be V-big divisors. We define the cycle
(L
1
- L2 - ...
E Ak(V)
- Lk)xEv
to be the maximum over all admissible models
#
: (X, 7) -+ (X, V) of
# (N1 . N2 - ....- N,_ .V)
where the Ni are nef and E := #*Li - Ni is a Q-divisor satisfying Ej ;>jp 0. In the
special case X = V, we write (L1 - L 2 -. .. - Lk)x.
If
#
(X, V)
-+
(X, V) is a smooth V-birational model, then
# (#*L1 - .... O*Lk)
#
p~f-- (L1 - ... - Lk)xiv.
Thus, if X is normal and V is any subvariety not contained in Sing(X), we define the
restricted positive product as the collection of classes (#*Li - ...
.
)gig as (X, V)
varies over smooth V-birational models.
Remark 2.4.7. As we consider all V-birational models #$: X -+ X, the Nip will be
big and nef, but not every big and nef divisor Hi < #*(Lilv) can be obtained in this
way. Thus we see that
( L1 . L2 -..-
Lk )xiv
_
( L1|v - L2|V-..-
LkIv)v
In general equality will not hold.
The restricted positive product satisfies a number of important properties.
Proposition 2.4.8 ([BFJ09],Proposition 4.6). The restricted positive product is symmetric, homogeneous of degree 1, super-additive in each variable, and continuous on
the p-fold product of the V-big cone.
Since the product is continuous, this allows us to define a limit as we approach
the pseudo-effective cone.
Definition 2.4.9. Let X be a normal variety, V a subvariety not contained in
Sing(X), and L 1 , L 2 ,... , Lk V-pseudo-effective divisors. Fix an ample divisor A on
X. We define the class
(L 1 - L 2
...
Lk)xiv-
lim((Li + EA)
e-o
.
(L 2 + eA)
(Ld+
....
eA))xlv.
Remark 2.4.10. It is easy to see that this definition is independent of the choice of
A. In fact, suppose that Bi is any sequence of V-big divisors converging to 0. Then
by sandwiching the Bi between suitably chosen ample divisors, we see that
lim ((L 1 + Bj) - (L 2 + Bj)..... (Ld + Bi))xv = (L1
.
L2
....
- Lk)xIv.
As a consequence, we see that even when the Li are only V-pseudo-effective the
restricted positive product still forms a compatible system under pushforward so that
our definition makes sense when X is singular.
In general the restricted positive product is only upper semi-continuous along the
V-pseudo-effective boundary.
Remark 2.4.11. [KM08] gives an alternative extension of the restricted positive
product to the pseudo-effective boundary. The main theorem of [KM08] shows that
for any two effective nef Q-divisors N 1 , N 2 there is some higher model # : Y -+ X so
that the coefficient-wise maximum max(#*N1, #*N2 ) is also nef without any bigness
hypotheses. Thus, we can define (L1 ... . Ld)* to be the maximum of pushforwards
of all products N 1 - ... - Nd on higher models, where the Ni <
#*Li
are effective nef
Q-divisors. When the Li are big this coincides with the positive product. It is unclear
whether the two coincide when the Li are only effective.
For our purposes, the most important case of the restricted positive product is
the following:
Proposition 2.4.12 ([ELM+09], Proposition 2.11). Let X be a normal variety, V
a d-dimensional subvariety not contained in Sing(X), and L a V-big divisor. Then
deg(Ld)xlv = volxIv(L).
We will need a few more properties of the restricted positive product. The first is
the observation that when the Li are nef, the restricted positive product reduces to
the usual intersection product.
Lemma 2.4.13. Let X be a projective variety, V a subvariety, and L1 ,..., Lk Vpseudo-effective divisors. Suppose N is a nef divisor. Then
(L1 . L2 - ... -Lk - N)xiv = {L1 - L2 - ... - Lk)XIV . N.
If H is a very general element of a basepointfree linear system, then
(L 1 - L 2 - ... - Lk)xIv - H = (Li - L 2 - ... - Lk)XlvnHProof. The first property is shown in [BFJ09], Proposition 4.7. To show the second,
consider a countable set of smooth V-birational models 4m :Xm -+ X such that the
restricted positive product can be computed using nef divisors on the Xm. Choose
H sufficiently general so that it does not contain any #-exceptional center. Then
the strict transform of V n H is a cycle representing the class #*H V. Thus we can
identify the classes
#rw,( N1 - N2 -..-
Nk
H ) = #mn,(N1
N2 -..-
= #m(N1 .N
2
Nk - $* H-V
Nk V nH)
...
The restricted positive product also has a natural compatibility with the divisorial
Zariski decomposition.
Proposition 2.4.14. Let X be a normal variety, V a subvariety not contained in
Sing(X), and L 1 ,..., Lk V-pseudo-effective divisors. Then
(L 1
...
- Lk)xIv =
(P,(L1 )
.
...
- P,(Lk)xiv.
Proof. First suppose that the Li are V-big. Since any nef divisor is movable, Proposition 2.2.12 shows that for any of the Ni as in Definition 2.4.6 we have P,(#*Li) ;>p Ni.
We also know that N,(#*Li) >pf #*N,(Li) since V is not contained in B_(Lj). Combining the two inequalities yields
#*P,(Li) ;>p~ Ni.
Thus the two classes are computed by taking a maximum over the same sets, showing
that they are equal.
Now suppose that the Li are only V-pseudo-effective. Fix an ample divisor A on
X. We will show that
(P,(L1 )....
P,(Lk))xIv =lim(P,(L1 +e A)
....
P,(Lk+ eA))xlv.
Combining this equality with the V-big case finishes the proof. Note that
Pa(L + cA) - P,(L) = cA + (N,(L) - N,(L + cA))
is V-big. Thus, our claim follows from Remark 2.4.10, which shows that if {Bj} is
any sequence of V-big divisors converging to 0 then
(P,(Li)
2.4.3
...
.
P,(Lk))xIv = lim ((P,(L1) + B )
(P,(La) + B 3 ))xiv.
...
Alternate Definitions
Since the restricted positive product is continuous, we can alter the definitions slightly,
Theorem 2.4.15 ([BFJ09], Proposition 2.13). Suppose that X is normal, V a subvariety not contained in Sing(X), and L 1, L 2 ,..., Lk are V-big divisors. Then the
following cycles in Ak(V) coincide:
1. (L1 . L 2 - ... . Lk)xIv.
2. The maximum over all V-birational models # X
$,( N1 - N2 . .-
where Ni is nef and
#*Li
-a
X of
-+
V
- Ni is V-pseudo-effective.
There is another characterization of the restricted positive product that emphasizes the relationship with the divisorial Zariski decomposition.
Theorem 2.4.16. Let X be a normal variety, V a subvariety not contained in
Sing(X), and L 1,..., Lk V-pseudo-effective divisors. Then
{L1j
where
.
Lk)xlv = inf $,(P,($*L1j9- ..
# : (X,V ) -+ (X, V)
P,($O*Lk)p)
ranges over all smooth V-birational models.
Remark 2.4.17. Since P,(#*L) #*P,(L), the classes on the right satisfy a filtering
property. Therefore the infimum on the right hand side exists.
Proof of Theorem 2.4.16: It suffices to consider the case when X and V are smooth.
Proposition 2.4.14 shows that the restricted positive product is invariant under passing to P,(Li). As remarked earlier, on any V-birational model # : (X, V) -+ (X, V)
(P,(#*L1) . ... - P,(#*Ld))J g -- (P,(#*L1)|g; . ... - P,(#*Ljpp)p'
This shows the inequality -. Conversely, recall that Proposition 2.2.15 shows that
(up to R-linear equivalence) there is a divisor G such that for any m there is a Vbirational model # : Y -+ X with
Nm,i
p P,(#*Li)
1
Nm,i + -$*,G
for some nef divisors Nmi. By definition
06(Nm,1 - . . -'Nm,d
V ) --- (L1j
Ld)|xiv.
Taking a limit as n increases proves the theorem.
2.5
D
Twisted Linear Series
It was observed by Iitaka that linear series of the form |[mL + A] Iplay an important
role in governing the numerical behavior of L. Due to the presence of the auxiliary
divisor A, we call these "twisted" linear series. Twisted linear series are much more
subtle than the classical case. There are a number of basic questions about such series
that remain open.
We begin by studying the asymptotic behavior of H0 (X, Ox (LmL + Aj)) as
m grows. This behavior is governed by a "twisted Jitaka dimension." Following
Nakayama, we denote this quantity by K,. We will see later that K, gives one of the
many equivalent notions of numerical dimension.
Definition 2.5.1. Let X be a normal variety, L a pseudo-effective R-divisor, and A
any divisor. If H 0 (X, Ox(LmL + A])) is non-zero for infinitely many values of m, we
define
,(L; A) := max k E Z>o lim sup hk(X, Ox([mL + A]))>
mk>}
The o--dimension r,, (X, L) is defined to be
m-oo
Otherwise we define r,(L; A)
-oc.
, (L) :=max f , (L; A)}
A
Since we are interested in perturbations by ample divisors, we will only consider
the case when A is a sufficiently ample Z-divisor.
Remark 2.5.2. Note that as we increase m the class of the divisor [mL] - [mL is
bounded. Thus if we replace [-j by [-] in Definition 2.5.1, the result is unchanged
as the difference can be absorbed by the divisor A.
The basic result about r, is the following:
Proposition 2.5.3 ([Nako4], V.1.4). Let X be smooth. Fix a very ample divisor H,
and suppose that A is an ample Z-divisor such that A - Kx - (n + 1)H is ample.
Then a divisor L is pseudo-effective iff h 0 (X, Ox(FmL] + A)) > 0 for every >> 0.
In the next section we will prove that K, is a birational invariant. Thus, K,(L) > 0
iff L is pseudo-effective. Our main theorem shows that K, coincides with the numerical
dimension, so we will see that all the properties of Theorem 2.0.4 hold for r,.
By analogy to the non-twisted case, one wonders if there is a polynomial in m of
degree K,(L) that is an upper bound on the rate of growth of H 0 (X, Ox([mL] + A).
Question 2.5.4. Suppose that L is pseudo-effective with K,(L) = k. Is it true that
for some sufficiently ample divisor A there are positive constants a and 3 such that
Om k
amk < h (X,Ox([mL + AJ))
for all sufficiently divisible m?
2.5.1
Restricted Twisted Linear Series
As usual, we will also be interested in a restricted notion that measures growth along
a subvariety V.
Definition 2.5.5. Let X be a normal variety, V a subvariety not contained in
0
Sing(X), L a pseudo-effective R-divisor, and A any divisor. If H (XIV, Ox(LmL +
A])) is non-zero for infinitely many values of m, we define
(
ho(X IV,0Ox(LmL + A)
h,(XIV,L;A):= max k E Z>o limmsup
Otherwise, we define K,(XIV, L; A) = -oc. The restricted o--dimension
is defined to be
K,(XIV, L):= max{s,(XIV, L; A)}.
0 }>
K,(XIV,
L)
A
Our first goal is to show that the restricted o--dimension is a birational invariant.
We will imitate the proof of [Nak04], V.2.7.
Proposition 2.5.6. Let X be a normal variety, V a subvariety not contained in
Sing(X), and L a pseudo-effective divisor. Suppose that # (X, V) -> (X, V) is a
V-birational model. Then
Ka (X
IV, L)
=
K,(XV#
)
Proof. For any ample Z-divisor A on X, there is an ample Z-divisor H on X with
H - #*A ample and effective. Thus we can naturally identify
H0 (X,Ox(LmLJ + A)) C Ho(X,O(Lm#5*Lj + H)).
Since there is an injection H 0 (V, Ov((LmLj + A)Iv) C H0(V, OI ((Lm#*LJ + H)ip)),
K,(XIV, #*L).
we see that ,.(XIV, L)
Conversely, let H be an ample divisor on X. Choose an ample divisor A on X such
that A y #,H. Note that #*[mL] - Lm#*LJ is effective and #-exceptional. Since
every #-exceptional divisor E satisfies E > 0, we have Lm#*L]+H p #*([mLl+A).
Thus
Ho (k|,
0k( Lm$* Lj + H) C H0 (XIV, O(#*( [mL] + A)).
If a section of Og(#*([mL] + A)) vanishes along V7, the corresponding section on X
vanishes along V. So
ho(X|V,0(L
*Lj + H)) < h0 (X|V,0x(FmL] + A))
and our conclusion follows from Remark 2.5.2.
Conjecturally, restricted K, coincides with the restricted numerical dimension. Unfortunately, I have only been able to show that vxiv(L) < n,(X|V, L). The following
question seems to be closely related to the reverse inequality.
Question 2.5.7. Let X be a smooth variety, V a subvariety of X, and L a V-pseudoeffective divisor. Is it true that
, (XX|V, L ) =
a (X
X|V, P, (L))?
Despite this gap in our knowledge, one can show that restricted K, satisfies many
properties similar to those of Ka.
2.5.2
Comparisons with Other Asymptotic Invariants
Our next goal is to relate restricted twisted linear series to our earlier constructions.
We first record an easy consequence of Nadel vanishing.
Lemma 2.5.8. Let X be a smooth variety and V be a smooth subvariety. Suppose
that L is a big Z-divisor and D = L is an effective divisor such that V is an isolated
component of the support of J(D). Then the restriction maps H 0 (X,Ox(Kx+L)) -+
H 0 (V, Ov(Kx + L)) are surjective.
The following proposition gives a lower bound on the dimension of the space of
restricted sections.
Proposition 2.5.9. Let X be a smooth variety, V a smooth subvariety, and L a
V-big divisor satisfying L >v 0. Then there is an ample divisor A depending only on
X such that for any V-birational model p : X -> X and any big and nef divisor N
satisfying 0 j, N < O*L after passing to a higher V-birational model X we have
H 0 (V,Op(N] + $*A)) C H0 (XIV,Ox(L] + A))
Proof. Since N 29 0, by passing to a higher V-birational model we may assume
that N has simple normal crossing support. We may also assume that Supp(N) has
transverse intersection with V.
Let f : Y -+ X be the blow-up of V with exceptional divisor F. Set k to be the
codimension of V so that Ky/x = (k - 1)F. Choose an ample divisor H on Y such
that H + kF is f-numerically trivial. We then choose an ample Z-divisor A1 such
that A1 - f, (H + kF) is ample. We choose ample A 2 such that A 2 - Kx is also ample.
Note that the map # X -+ X may have some exceptional centers contained in
V. Let E 1 , . . . , Ek be the exceptional divisors above such centers. Choose a, so that
VE (0*A 1 - aiEs) = 0. Each ai is an integer no larger than k. We define E = T ajEj.
Note that E < Kk/X since every exceptional center contained in V has codimension
at least k + 1.
The next step is a multiplier ideal calculation. Let @ : W -+ X be a log resolution
of Supp(N)USupp(#*A1)Uexc(#). By passing to a higher model, we may assume that
there is a map g : W -* Y that is a composition of blow-ups along smooth centers.
We let Fw denote the strict transform of F. There is a divisor Aw - @*#*A1 such
that Aw 2 @*E and [Aw] = kg*F. Now, [Aw] - @)*E only has positive coefficients
over O-exceptional divisors. In particular, Kw/R - [Aw] + @*E > -Fw. Since the
coefficients of 0*[N] - @*N over O-exceptional divisors are fractional, we find that
there is some divisor M = [N] + *A1 - E with J(M) = Ig.
Applying Lemma 2.5.8 to [N] + (#*A1 - E) + #*(A 2 - Kx), we see that the
restriction map
HO (k, 0(K,
+ [N] + (#* A - E) + #*(A2 - Kx)))
-+ H (V, Og (K
+ [Ni + (#*A1 - E) + #*(A 2
is surjective. Since Kklx - E is effective and
Thus for any divisor D on X we have
H0 (XIV, 0x(D)) = H (|iV,
#-exceptional,
Og(D+ K
-
Kx)
we have Kj 1j - E
x
-
p 0.
E).
Similarly, note that [N] g [#*L] p #*[L]. Since K+[N]+#*A1+#*(A2-Kx)
[N] + #*(A1 + A 2 ) + (KJ1I - E), we obtain
H (, Og ([N + #*(A1 + A 2 )))
C H (V, Og([N] +# *(A1 + A2 )+ K~1l - E))
H0 (XIV, Ok([N] +# *(A + A 2 )+ K/x - E))
c H (XIV, 0($*[L] + $*(A A+ A 2 ) + Kkx - E))
=H (X|V,0x( [L] + A1 + A 2 )).
Setting A = A1 + A 2 finishes the proof.
2.6
Nakayama Constants
Suppose that L is an ample divisor and V a subvariety in X. Let # Y -+ X be
a resolution of the ideal Iv, and define the divisor E by the equation Oy(-E) =
#41v
- Oy. The Seshadri constant
E(L, V) := max{ F I#*L - rE is nef }
measures "how ample" L is along the subvariety V. Seshadri constants play an
important role in understanding the positivity properties of ample divisors. There is
a similar notion that can be defined for any V-big divisor L: the moving Seshadri
constant is
E(||L||, V) := max E(A, V)
#*L;>pA
where
(X,
k: V) --> (X, V) varies over all V-birational models.
We will be interested in a related notion that can be defined for an arbitrary
pseudo-effective divisor L.
Definition 2.6.1. Let I be an ideal sheaf on X and let L be a pseudo-effective divisor.
Choose a resolution q : Y -> X of I and define E by setting Oy(-E) = #-1 - Oy.
We define the Nakayama constant
s(L, I) := max{ r
L - TE is pseudo-effective }.
Of course, s is independent of the choice of resolution. When I is the ideal sheaf of
a subvariety V, we will also denote the Nakyama constant by s(L, V).
Note that s(L, V) can be positive even when L is only pseudo-effective. This is
in contrast to the movable Seshadri constant, which vanishes as we approach the
pseudo-effective boundary. In a sense, the movable Seshadri constant measures the
ampleness of L along V, whereas the Nakayama constant measures the bigness of L
along V. Thus, the Nakayama constant shares a closer relationship with the other
invariants we have considered.
There is a useful criterion which is closer in spirit to Nakayama's original formulation.
Proposition 2.6.2. Let X be a projective variety, V a subvariety, and L a pseudoeffective divisor. Then ;(L, V) > 0 iff there is some sufficiently ample divisor A on
X so that for any q,
h (X,I @9Ox((mL + Aj)) > 0
for sufficiently large m. As (q, m) varies over all pairs of positive integers satisfying
the above criterion we have 4(L, V) = lim supq, max,
.
Proof. Let # : Y -+ X denote a smooth resolution of the ideal sheaf of V and define
E by Oy(-E) = #rIy - Oy. Note that 4*L - TE is pseudo-effective for some T > 0
iff for any b, there is some a so that a#*L - bE is pseudo-effective. By Proposition
2.5.3 (and Remark 2.5.2), this can be true iff for every b there is some a such that
h0 (Y, Oy(Lc(a#*L - bE)j + H)) > 0
for all c and for some fixed ample divisor H. Choose ample A ;> #H. Replacing
[ac) by rn and [cb] by q, we see that it is equivalent to require that for any q
h (X,I; ® Ox(LmLj + A)) > 0
for sufficiently large m. The last statement follows by comparing coefficients.
0
This allows us to show that the Nakayama constant behaves well with respect to
V-birational models. We first need a lemma.
Lemma 2.6.3. Let X be a projective variety and V be a subvariety of X. Suppose
that # (X, V) -+ (X, V) is a V-birational model. Then #,IV = Iq for all sufficiently
large q.
Ox has image contained in
Proof. It is clear that the injection #,IV -+ #0 _
Thus, it suffices to show that the map #,,I -+ Iq is surjective. Since V is a
I.
Op. Thus, it suffices to show that the map
D #
subvariety, we know that
#. (#-I-v -Ox)
-+
IS.
is surjective for sufficiently large q.
Note that if : X -+ X is a higher model, then
((
0 0 II
o
)q C (#-Iv - 02) -
So, it suffices to prove the statement for a sufficiently high model. In particular,
we may assume that X admits a morphism @ X -+ BlvX. Let E denote the
Cartier divisor defined by the inverse image ideal sheaf on the blow-up of V. Then
# Iv - OO = 02(-O*E) and 'i),(#- 1 v - O2) = 0(-E). Thus, we have reduced to
checking whether
f,0BlvX (-qE
)
_TV'.
where f : BlvX -+ X is the blow-up along V. However, since -E is the relative
twisting sheaf 0(1) we know that
f*f,0BlvX(-qE)
-+
OBx(-qE)
is surjective for sufficiently large q, finishing the proof.
E
Proposition 2.6.4. Let X be a normal variety, V a subvariety, and L a divisor. If
(X, V) is a V-birational model for (X, V), then ,(#*L, V) =§(L, V).
Proof. We will use Proposition 2.6.2. First, note that for any ample A on X there
1
- OP .
is an ample H on X with H ;>i #*A. Furthermore we have I D->I2
VVi
Thus for any (q, m) satisfying h(X,4 09 Ox(LmL] + A)) > 0 we have h (X, I ®
Ok ([m#*L] + H)) > 0. In particular s(L, V) < s(#*L, V).
Conversely, fix an ample H on X and suppose that (q, m) satisfies ha(X,oI
O(Ln*L] + H)) > 0. For any ample A > #,H we have 0#*A >, #*#,H >p H.
the push-pull formula, the previous lemma indicates that h (X, Iv 0Ox( [mL +A))
0 when q is sufficiently large.
0
By
>
0
The following proposition indicates that the Nakayama constant satisfies the usual
compatibility relations.
Proposition 2.6.5. Let X be a normal variety, V a subvariety not contained in
Sing(X), and L a V-big divisor. Then
s(L,V) =(P,(L), V) = max g(A,#
p*L>A
where
#
Y
-
Iv - Oy).
X varies over all birational maps.
Proof. We start with the equality on the left. It suffices to show the inequality <.
Furthermore, by passing to a V-birational model we may assume that X and V are
smooth. Let # : Y -+ X denote the blow-up of V and let E denote the exceptional
divisor. Suppose that #*L - rE is pseudo-effective. Fix an ample A on Y. For any
E > 0, we find that #*L + EA ~ TB
E + F for some effective F. Since Supp(E) is not
contained in the restricted base locus, we know that N,(#*L + eA) < F. Subtracting,
we find that P,(#*L + EA) - TE is pseudo-effective. Taking a limit over e and noting
that #*P,(L) > P,(#*L) completes the inequality.
We now turn to the equality on the right. Since § can be computed on any
model, the inequality > is a result of the fact that any A < #*L satisfies A <
P,(#*L) < *P,(L). Conversely, fix an ample divisor H on X. Theorem 2.2.15
indicates that there are birational maps 4m and big and nef divisors N, satisfying
N_ 5 P(#*L) 5 Nm + -#* H. Choosing ample divisors near the N., we see that
the expression on the right hand side can be made arbitrarily close to s(P,(L), V). E
Finally, we remark in passing that [Nak04] shows that g(L, V) is controlled by
what happens to a very general subvariety of dimension equal to dim(V).
Proposition 2.6.6 ([Nak04], V.2.21). Let V be a d-dimensional subvariety of a
smooth variety X and let L be a pseudo-effective divisor. If 4(L, V) = 0, then for
sufficiently general ample divisors H 1, . . . , H,_a we have §(L, H1 n ... n Hn-ad) = 0.
2.6.1
Restricted Nakayama Constants
Suppose that V is a subvariety of X and that W is a subvariety of V. We will define
the restricted Nakayama constant by considering the positivity of (L)xv along W.
Definition 2.6.7. Let X be a normal variety, V a subvariety not contained in
Sing(X), and W a subvariety of V not contained in Sing(X) or Sing(V). Suppose
that L is a V-pseudo-effective divisor. We define the restricted Nakayama constant
CxIv(L, W) as follows.
Let # X -> X be a W-birational model such that V and W are both smooth.
We can identify (#*L)kig as a rational equivalence class of codimension 1 cycles on
V. Since V is smooth, this corresponds to an R-linear equivalence class of R-Cartier
divisors. Since ; only depends on numerical classes, it makes sense to define
;xlv(L, W) := sp((#*L) Ig,W).
This definition is independent of the choice of model since if / : X -> X is another
smooth W-birational model then
cp(#*Lgig W)=
yg($~/*(#*L>ig, W)
=P gg($#L)jI), W)
Although it is not immediately clear that the restricted Nakayama constant reduces to our earlier definition when V = X, Proposition 2.6.4 shows the consistency
of the two definitions since gxlx(L, V) = sx(P,(L), V).
Proposition 2.6.8. Let X be smooth, V a smooth subvariety of X, and W a subvariety of V. Suppose that L is a V-big divisor. Then
Cxiv(L, W) = max cf,(Ajp,
4*L>p A
as
4: (X, ) -+
# 1 Iw -0k)
(X, V) varies over all V-birational models.
Proof. Since V is smooth we know that Cxjv(L, W) = §v((L)xiv, W). By definition the classes of #,Alp converge to (L)xiv. Thus we can conclude by applying
Proposition 2.6.5.
We would like to have a criterion for non-vanishing of the Nakayama constant
in terms of sections. Unfortunately, I have been unable to prove an analogue of
Proposition 2.6.2 for the restricted Nakayama constant.
Question 2.6.9. Let X be a normal variety, V a subvariety not contained in Sing(X),
and W a subvariety of V. Suppose L is a W-pseudo-effective divisor. If there is some
sufficiently ample divisor A on X so that for any q
ho(XIV, Iw/X 0 Ox(LmL + AJ)) > 0
for sufficiently large m, can we deduce that gxlv(L, W) > 0?
This question is closely related to Question 2.5.7.
2.7
The Restricted Numerical Dimension
We now turn to the numerical dimension. Recall that for a nef divisor L, the numerical
dimension measures the maximal dimension of a subvariety W C X such that L is
big along W. It is precisely this intuition that we want to keep in extending our
definition to a general pseudo-effective divisor. There have been several proposals of
such a definition in [Nak04) and [BDPP04]. The common feature of all of these is
that we should not consider the restriction Liw but a "positive restriction" of L. Our
goal in this section is to show that these definitions coincide.
We will work with a restricted version throughout. The restricted numerical dimension of L along V measures the maximal dimension of a general subvariety W c V
such that the "positive restriction" of L is big along W.
Theorem 2.7.1. Let X be a normal variety, V a subvariety not contained in Sing(X),
and L a V-pseudo-effective divisor. In the following, W will denote an intersection of
very general very ample divisors with V, and A some fixed sufficiently ample divisor.
Then the following quantities coincide:
1. max{k G Z;>oI(Lk)xyv
0}.
2. max{dim W|(Limv)xIw > 0}.
3. max{dim WI lime-o volxIw(L + EA) > 0}.
4. maxfdim WIlim info volf (P,(#*L)|i)
> 0} where
: (X, W) --+ (X, W)
ranges over W-birational models.
5. max{dimW|(L)xiw is big }.
6. min{dimWcxiv(L, W)= 0}.
By convention, if L is big we interpret this expression as returningdim(V).
This common quantity is known as the restricted numerical dimension of L, and is
denoted vxiv(L). It only depends on the numerical class of L.
Furthermore, Vxiv(L) is no greater than
7. max {k E Z>o limsup
eh
(XI V,
Ll +A)
We have equality in the special case X = V. If we assume an affirmative answer to
Question 2.6.9, then equality holds in general.
(2) is the definition of numerical dimension in [BDPPO4], while (6) and (7) correspond to r,(L) and r,(L) in [Nak04] respectively.
We will prove Theorem 2.7.1 using a cycle of weak inequalities. Much of the work
has been done already in previous propositions.
Proof. We first note that each of these quantities is invariant under passing to a
smooth V-birational model. This is implicit in definitions of the characterizations
(1), (2), (4), and (5). To see the invariance for (6), recall that by definition we
first pass to a smooth V-birational model and take the strict transform of W. By
Proposition 2.6.6, we may replace W by an intersection of very general very ample
divisors on our smooth model. The invariance is shown for (7) in Proposition 2.5.6.
To show the invariance for (3), suppose that A is an ample divisor on X. For any
smooth V-birational model # : X -+ X, the pullback #*A is V-big. By sandwiching
#*A between suitable ample divisors on X, we see that the limit defined in (3) is the
same on X and X. Thus we will assume throughout that X and V are smooth.
Let H 1 ,..., Hd-k represent very general elements of a very ample
- i . . Hd-k > 0.
Proposition 2.4.13 allows us to conclude that (Lk)xlVnHln...nHd k > 0.
(1) < (2).
linear system. Statement (1) is equivalent to the fact that (Lk)xiv
(2) K (3). Proposition 2.4.12 shows that conditions set on W in (2) and (3) are
the same.
(3) < (4). By Proposition 2.4.16 we know that
volxIw(L + EA) =
vol- (P,(#*(L + cA))|
inf
)
4:x->X
where # (X, W) - (X, W) varies over V-birational models. Since volume varies
continuously on every model, taking the limit as E goes to 0 proves that the two are
equal.
(4) < (5). By assumption there is a positive constant a that gives a lower bound
for the volume of the divisor P,(#*L)|i; where # : (X, W) -+ (X, W) is any Wbirational model. Consider the pushforward of these divisors under #,. The volume
does not decrease under pushforward, so a is also a lower bound for the volume of
the pushforwards on W. By Proposition 2.4.16 (L)w is the limit of the classes of
#,,P,(L)|g, so it must also be big.
(5) K (6). We defer this inequality until later.
(6) K (1). We need to consider the 0-case separately. Note that (1) is 0 precisely
when (L)xiv is numerically trivial. This means that (6) is also 0. Thus, we can prove
that (6) (1) by considering the case where (6) is at least 2 and (1) is at least 1.
Set k to be the value of (1), and suppose that k is less than the value of (6).
Let W be a k-dimensional intersection of V with general very ample divisors. Set
T = CxIw(L, W) > 0, and let # : Y -+ X be the blow-up of W where E is the
exceptional divisor.
Fix a very ample divisor H on Y. We will start by analyzing #*L + EH for
Ai,, is a sequence of
small e. Suppose that @i,, : Yi,, -+ Y and @i,,(#*L + cH) ;>
V-birational Fujita approximations whose limit computes ((#*L + CH)d+1),Ig. By
Proposition 2.6.8, we know that Aj,, - L*Ej is pseudo-effective for i sufficiently
large. Thus
0 <
E
V)p*
Ai,e -
H-k1
- V .AA-
By pushing forward and taking the limit over i, we find
0
Hd-k-i
H)k+l),
((#*L +H
-
((#L +eH))y
- E - H dk-1.
This is true for all sufficiently small e, so
0
(*Lk+
1y
.
H -k--
-
T(
k)
-E
- Hd-k-1.
By assumption (#*Lk) yp -# 0. By choosing sufficiently general elements H 1 , .. . , Hd-k_ 1 E
|HI, we may ensure that E nH 1 n... n Hdk_1 n 7 maps finitely onto W via #. Since
W is sufficiently general, we have (#*Lk)yp- E - H dk-1 > 0. Thus we obtain
0 < ($*Lk+l)xIv - H d-k-1
contradicting the fact that (Lk+l)xIv = 0.
This completes all of the comparisons except (5) < (6). We now turn to the
quantity (7).
(5) < (7). Choose an ample divisor A as in Theorem 2.1.13. Thus for every
positive integer m there is a divisor D, - [mLl + A such that Dm >v 0. For any m
there is a sequence of W-birational models #j,m : Xj,m -+ X centered in B_ (L) and
big and nef divisors Nj,m on Xj,m such that Nj,m
converge to (Dmz)xiw -R m(L + !A')xlw.
some ample A such that
H0 (I%-,m, OV
W,m
#m(Dm)
and
#j,m,(Nj,mI~,m)
By Proposition 2.5.9, we know there is
([Nj,ml + #*A)) C H0 (XIW, Ox( [mLI + A' + A))
Since H0 (XIV, Ox(FmL + (A' + A))) is at least this large, it suffices to show that
the spaces H0 (Wj,m, 0 iy' ([Nj,ml + #*A)) grow at the necessary rate.
Choose a representative B of (L)xiw satisfying B > 0. Proposition 2.2.15 indicates that for any effective ample divisor G there are models )m : Wm -> W and big
and nef divisors Mm on Wm such that Alm 5 mP,(* B) 5 Mm, + #*G. Comparing
against the Nj,m, we see that by replacing A by A + G we get the necessary rate of
growth.
(7) < (6). Nakayama proves this in the case X = V. If we assume a positive
answer to Question 2.6.9, it should be possible to mimic his argument to prove it for
general V.
Finally, we return to (5) < (6). We have proved this in the case when X = V and
L is a divisor on X. However, the inequality for general V is precisely this situation
for the variety V and the divisor (L)xlv on V. This finishes the equality of the first
six characterizations, concluding the proof.
The restricted numerical dimension is very natural from the viewpoint of birational
geometry. It satisfies a number of important properties.
Theorem 2.7.2. Let X be a smooth variety, V a subvariety of X, and L a V-pseudoeffective divisor.
1. If L' is also V-pseudo-effective, then uxlv(L + L') > vxlv(L).
2. When L is nef, vxlv(L) = vv(Llv).
3. We have 0 < vxlv(L)
dim(V).
4. vxlv(L) = dim(V) iff (L)xjv is big and vxlv(L) = 0 iff (L)xjv is numerically
trivial.
5. If # : (,X ,V ) -+ (X,V) is a V-birational model then uv1 Q($*L) = vxlv(L).
6. We have vxiv(L) = vxiv(Pa(L)).
Proof.
1. By definition (L + L')xiw >- (L)xiw. So this inequality follows from characterization (5) of the restricted numerical dimension in Theorem 2.7.1.
2. The restricted volume of an ample divisor can be calculated as an intersection
product, so the equality follows from characterization (3) in Theorem 2.7.1.
3. This set of inequalities is obvious.
4. This follows from characterizations (1) and (5) in Theorem 2.7.1.
5. We have checked this already in proving Theorem 2.7.1.
6. This follows from the fact that the restricted positive product is invariant under
passing to P, as demonstrated in Proposition 2.4.14.
Remark 2.7.3. It is important to note that we can have vxlv(L) = dim(V) even
when L is not V-big. This is demonstrated in Example 2.3.8.
In the case when X = V, [Nak04] shows several additional properties.
Proposition 2.7.4 ([Nak04],V.2.7). Let X be smooth, L a pseudo-effective divisor.
1. We have ri(X, L)
2. Suppose that
f
v(L).
: X -- Z has connected fibers, and F is a very general fiber.
Then v(L) < v((L|F) + dim(Z).
3. Iff : Y -+ X is a surjective morphism then v(f*L) = v(L).
We now recall the compatibility relations mentioned in the introduction. They all
follow immediately from the definitions.
Proposition 2.7.5. Let V be a normal subvariety of X and L a V-pseudo-effective
divisor. Then
1. vxlv(L) < v(L).
2. vxlv(L)
v(L v).
3. If H is a very ample divisor and vxiv(L) < dimV, then vxlvnH(L)
vxlv(L).
It is interesting to note that v is not lower semicontinuous as might be expected.
This is a consequence of the fact that the restricted positive product is only upper
semicontinuous on the boundary of the V-pseudo-effective cone.
Example 2.7.6 ([BFJ09], Example 3.8). Let X be any smooth surface with infinitely
many -1 curves. Take some compact slice of NE(X). We can choose a convergent
sequence of distinct classes {ai} on this compact slice such that each a, lies on a ray
generated by a -1 curve. Note that for any irreducible curve C there is at most one
i for which a, . C < 0. Thus # := limi_.oo a2 must be a nef class. Since -1 curves
are contractible, we find that v(ai) = 0 for every i. However, a non-trivial nef class
# has v(0) > 1. Thus v is not lower semicontinuous.
Question 2.7.7. What properties does v satisfy along the V-pseudo-effective boundary?
Chapter 3
Pseudo-effective Reduction Maps
Suppose that X is a smooth complex projective variety and L an effective Cartier
divisor on X. The Iitaka fibration #L associated to L is a fundamental object of study
in birational geometry. However, it is often very difficult to understand explicitly. Our
goal in this chapter is to understand to what extent properties of #L are determined
by the numerical class [L] E N 1 (X). More precisely, we will construct a rational map,
depending only on [L], which approximates the Iitaka fibration associated to L. We
are primarily interested in divisors L whose numerical class is on the boundary of the
pseudo-effective cone.
The main interest in such results arises from the Abundance Conjecture, one of
the key components of the birational classification of varieties. Loosely speaking,
the conjecture states that the Iitaka fibration of the canonical divisor Kx is precisely
determined by numerical properties. We will show later on that the general framework
laid out here provides additional insights in the special case of the canonical divisor.
Since the Iitaka fibration #L for L is characterized by the fact that i(F,LIF) = 0
for a very general fiber F, one might hope to construct a numerical approximation for
#Lby requiring that LIF is numerically trivial for a general fiber F. In [BCE+02], the
authors construct such a map when L is a nef divisor. The main result is that there
is a rational map f : X -- + Z with LIF=-0 that is maximal in the sense that any
other such fibration factors birationally through it. The authors call this fibration
the nef reduction map.
We would like to construct a similar map for an arbitrary pseudo-effective divisor
L. It turns out that numerical triviality is no longer the right condition to impose on
the fibers. Rather, we should require that the numerical dimension v(LIF) vanishes,
which is a less restrictive condition when L is not nef. Our goal is to demonstrate
the following theorem of [Eck05]:
Theorem 3.0.8 ([Eck05], Proposition 1.5 and Definition 4.1). Let X be a smooth
projective variety and L a pseudo-effective R-divisor on X. There is a birational
model # : Y -+ X and a morphism r : Y -+ Z with connected fibers satisfying
1. For a general fiber F of r, we have v(L|F) = 0.
2. The pair (Y, 7r) is the maximal quotient satisfying (1): if $' : Y' -+ X is a
birational map and r' : Y' -+ Z' a morphism with connected fibers that satisfies
(1), then there is a dominant rational map $ : Z' -- + Z such that 7r = @ r' as
rational maps.
The pair (Y, 7r) is determined up to birational equivalence and depends only on the
numerical class of L. We call it the pseudo-effective reduction map associated to L.
Since [Eck05] obtains this result from an analytic perspective, the main contribution of this chapter is to recover and extend the work of [Eck05] using algebraic
techniques. We will discuss the relationship with Eckl's work in Section 3.2.3. Our
approach yields the following description of the pseudo-effective reduction map.
Theorem 3.0.9. Let X be a smooth projective variety and L a pseudo-effective Rdivisor on X. The pseudo-effective reduction map for L is the generic quotient of X
by all movable curves C satisfying vx 1c(L) = 0.
The pseudo-effective reduction map has the following important properties:
Theorem 3.0.10. Let X be a smooth variety and let L be a pseudo-effective divisor.
1. Suppose that p : Y -> X is a birational map. The pseudo-effective reduction
map for #*L is birationally equivalent to the pseudo-effective reduction map for
L.
2. If K(X, L) > 0 then the Iitaka fibration for L factors birationally through the
pseudo-effective reduction map.
3. The pseudo-effective reduction map for P,(L) is birationally equivalent to the
pseudo-effective reduction map for L.
4. If rc(X, L) > 0, there is a birational model # : W -+ X and a morphism f
W - Z birationally equivalent to the pseudo-effective reduction map such that
there is a divisor D on Z and an effective divisor E on W with p*L ~( f*D+E
and the section rings R(X, L) = R(Z, D) coincide.
A pseudo-effective divisor L is said to be abundant if K(X, L) = v(L). It is wellknown that abundant nef divisors have many special geometric properties. Using the
pseudo-effective reduction map we will show that many of these properties generalize
to the pseudo-effective case. The importance of abundance derives from the following
reformulation of the Abundance Conjecture:
Conjecture 3.7.3. Let (X, A) be a kit pair. Then Kx + A is abundant.
The pseudo-effective reduction map naturally leads to an inductive approach to
the Abundance Conjecture. Using the work of [Amb04], we show the following result:
Theorem 3.0.11. Let (X, A) be a klt pair. Assume that Kx + A is pseudo-effective,
and let f : X -- + Z denote the pseudo-effective reduction map. If Conjecture 3.7.3
holds on Z, then Kx + A is abundant.
In this approach to the Abundance Conjecture, the key question is whether the
pseudo-effective reduction map for Kx+A maps to a variety of smaller dimension. By
Theorem 3.0.9, this question can be answered by finding curves satisfying a numerical
condition. Similar work has appeared in the recent preprint [Siu09).
3.1
Background
In this section, we will review the basic tools needed in this chapter. We begin with
a brief discussion of several basic notions in birational geometry. We then discuss the
quotient theory of [Cam81] and [KMM92], which describes how to define a quotient
by families of subvarieties.
We will continue to work with over the base field C. All varieties are assumed to
be projective unless otherwise qualified. Just as before, the term "divisor" means an
R-Cartier divisor unless otherwise qualified.
3.1.1
Birational Geometry
We will need the following standard terminology for birational maps.
Definition 3.1.1. Suppose that -r : X -- + Z is a rational map and let U c X be
the locus on which 7r is defined. Then we say that 7r is almost proper if the general
fiber of 7r : U -+ Z is proper. It is also common to say that 7r is almost holomorphic,
but we will not use this terminology.
Definition 3.1.2. Suppose that r : X -- + Z and 7r' : X' -- + Z' are two rational
maps. We say that 7r and -r' are birationally equivalent if there are birational maps
#: X -- + X' and yP: Z -- + Z' such that - o ir = ro # on an open subset of X.
Definition 3.1.3. Suppose that 7r : X -- + Z is a rational map. We say that 7 is a
birationalcontraction if some resolution (and hence any resolution) of ir has connected
fibers.
We also take this opportunity to recall a special case of Hironaka's flattening
theorem.
Theorem 3.1.4 ([Hir75]). Let f : Y -+ X be a surjective morphism of (integral)
projective varieties. There is a birational morphism $ : X' -+ X satisfying the
following: let Y' denote the unique component of X' xx Y that dominates Y under
the projection map. Then the natural morphism v : Y' - X' is flat.
We may also require that the X' constructed in the theorem is smooth.
3.1.2
Relations and Quotients
In this section we revisit the quotient theory developed by Campana in [Cam8l] and
Kollir, Miyaoka, and Mori in [KMM92] (and further developed in other papers). The
presentation below can be found in [Kol96], Section IV.4.
Let X be a variety. Suppose that we are given a family of subvarieties of X, that
is, a family of algebraic varieties s : U -+ V admitting a morphism w : U -+ X. We
would like to construct a "quotient" h : X -- + Z by this family of subvarieties. At
the very least, we should ask that a general element of our family be contracted to a
point by h. In other words, any two general points connected by a chain of fibers of
s should be identified under h. In light of this requirement, we define an equivalence
relation on X by saying that two points are equivalent if they can be connected by
such a chain. Surprisingly, it is possible to quotient out by this equivalence relation
after setting only a few conditions on the morphisms s, w.
We now make the construction more precise. Let (U, V, s, w) denote the family
of subvarieties defined by the morphisms s : U -+ V and w : U -> X. We will
allow U and V to be quasi-projective varieties, so that our family is not necessarily
proper. Rather than dealing with just a single family, we allow finitely many families
(Ui, V, si, wi) where i ranges from 1 to k. Given such data, [Kol96] defines a constructible subset
(U1 , ...
, Uk) C X x X such that two points x 1 , X2 can be connected
. , Uk).
by a chain of subvarieties iff (X1 , X2 ) E (U 1 , ..
There are two basic versions of the quotient theorem, and we will use both of
them. The first version deals with open morphisms.
Theorem 3.1.5 ([Kol96],IV.4.13). Let X be a normal variety and (Us, Vi, si, wi) be
finitely many families of quasi-projective subvarieties on X. Suppose that each wi
is open and each si is flat with irreducible fibers. Then there is an open subvariety
X 0 C X and a morphism i : X0 -+ Z 0 with connected fibers such that
1. (U1,... ,Urn) restricts to an equivalence relation on X 0 .
2. For every z E Z0 the closure of 7- 1(z) in X coincides with the closure of a
suitable (U 1 , . . . ,U)-equivalence class.
3. For every z E ZC two generalpoints of ir (z) can be connected by a (U 1, ... ,U) chain of length at most dim X 0 - dim Z0 .
This theorem shows that we can construct a quotient when we take into account
only the "generic" connections. Using this theorem inductively, one obtains the (more
common) version that deals with proper morphisms.
Theorem 3.1.6 ([Kol96],IV.4.16). Let X be a normal variety and (Ui, Vi, si,wi) be
finitely many families of subvarieties on X. Suppose that each wi is proper and that
each si is proper with connected fibers. Then there is an open subvariety XC C X, a
proper morphism h : X 0 -> Z, and an open subset Z' C Z such that
1. (U 1,... ,U) restricts to an equivalence relation on X 0 .
2. For every z E Z 0 the fiber 7-1 (z) coincides with a (U1 ,...,Um)-equivalence
class.
3. For every z E ZO two points of i--1(z) can be connected by a (U1, ... ,Urn)-chain
of length at most 2 dinX-dirnZ - 1.
There are also versions for infinite collections of families.
3.2
Previous Work
The concept of a reduction map associated to a pseudo-effective divisor has its origin
in a preprint by H. Tsuji. Loosely speaking, given a variety X and a pseudo-effective
divisor L, we would like to describe the maximal quotient r : X -- + Z that is "trivial"
with respect to L in some sense. In the first two subsections, we will review [BCE+02]
and [BDPP04], which quotient out by curves with L - C = 0. In the third, we review
[Eck05] which constructs a pseudo-effective reduction map using analytic techniques.
3.2.1
Nef Reduction
In [BCE+02] the authors construct a reduction map for nef divisors. They note that
when L is nef one can obtain a good theory by "quotienting" by all the curves C with
L - C = 0.
Theorem 3.2.1 ([BCE+02], Theorem 2.1). Suppose that X is normal and L is a nef
divisor. There exists an almost proper dominant map F : X -- + Z with connected
fibers such that
1. L|F
=
0 for every proper fiber F of dimension dim X - dim Z.
2. The map 7r is the maximal quotient satisfying (1): if 7' : X -- + Z' is another
almost proper dominant map satisfying (1), then there is a dominant rational
map p : Z' -- + Z such that 7 = @o 7r' as rational maps.
This map
f
is unique up to birational equivalence.
The map 7r is constructed by applying Theorem 3.1.6 to all families of L-trivial
curves. Property 2 follows automatically; the main difficulty is assigning a geometric
meaning to the fibers of 7r. Theorem 3.1.6 guarantees that a general fiber F admits a
connecting family of L-trivial curves, but it is not immediately clear that this implies
anything about the restriction of L to F. The connection to the geometry is given
by the following theorem.
Theorem 3.2.2 ([BCE+02], Theorem 2.4). Let X be an irreducible projective variety
and L a nef divisor on X. If any two points of X can be joined by a chain of irreducible
curves with L - C = 0, then L = 0.
3.2.2
Quotienting by Movable Curves
In [BDPP04] the authors consider how to generalize this work to arbitrary pseudoeffective divisors. Note that a rational map 7 : X -- + Z is characterized up to
birational equivalence by the movable curves that it contracts. In other words, we
do not lose anything by restricting our attention to movable curves, and it turns out
that this constraint is well-suited for the study of a general pseudo-effective divisor
L.
The authors of [BDPP04] define a reduction map by applying Theorem 3.1.6
to all families of movable curves C with L - C = 0. We obtain an almost proper
rational map 7r : X -- + Z, and again the main problem is to assign a geometric
meaning to the fibers. The difficulty is that the fibers of this map do not have a nice
interpretation. The following example shows that this map may contract more than
the Iitaka fibration.
Example 3.2.3. ([BDPPO4], Example 8.9) We construct an example of a smooth
threefold X, a divisor L on X with s(X, L) = 1, and a connecting family of movable
curves C such that L - C = 0.
Let X' be the smooth projective bundle
p: P(O e 0 O(-1)) --> P1 .
Each fiber of p is isomorphic to P2 . We define the divisor F to be a fiber of p.
We let T' denote the section P(O(-1)). Note that T' has normal bundle NT,/x, =
O(-1) EO(-1). Thus, we can construct the flop 4: X' -- + X of T'. More explicitly,
let 7r : Y -+ X' denote the blow up of X' along T', and let E '_ Pl x P 1 denote the
exceptional divisor. We can contract E along either projection; one returns the map
to X', the other defines a new smooth variety X. We define the divisor L on X to be
the strict transform of a fiber F on X'. The pair (X, L) will provide our example.
We first construct a connecting family of curves with L . C = 0. Choose any line
C' in F that avoids T' and let C denote the strict transform of this line. Since # is
an isomorphism on a neighborhood of C, we have L - C = F - C' = 0. We claim that
deformations of C define a connecting family of curves on X. Any two points in a
strict transform of a fiber of p can be connected by deformations of C. Furthermore,
when we deform C so that it intersects the flipped curve T, then C breaks up into
two components, one of which is T itself. So, we can traverse between the strict
transforms of two different fibers of p via the flipped curve T, showing that any point
can be connected to any other.
However, K(X, L) = K(Y, 7r*F + E) = 1. Thus, the quotient of the family defined
by C contracts more than the Iitaka fibration.
Nevertheless, in some cases they are able to show that connection properties translate to geometric properties.
Definition 3.2.4. Let X be smooth and L be a pseudo-effective divisor. Suppose
that Ci are movable curves on X with L - Ci = 0. We say that the Ci are strongly
connecting if two general points of X can be connected by deformations of the C
such that L -T = 0 for every component T of the connecting chain.
By generalizing the ideas of [BCE+02] the authors prove
Theorem 3.2.5 ([BDPP04], Theorem 8.7). Let X be a smooth variety and L be a
pseudo-effective divisor. Suppose that Ci is a strongly connecting collection of movable
curves for L. Then v(L) = 0.
3.2.3
Analytic Approach
Eckl has further developed the notion of a reduction map in a series of papers [Eck04a],
[Eck04b], [Eck05]. In the first two, he works out a correct formulation of Tsuji's
original ideas concerning multiplier ideals. In the third, he constructs a pseudoeffective reduction map. Since our goal is to give another construction of this map,
we will briefly recall the ideas in this paper.
Suppose that X is a compact Kahler manifold with Kihler class w. We first specify
when a (not necessarily algebraic) foliation F is numerically trivial with respect to a
pseudo-effective (1, 1)-form a.
Definition 3.2.6 ([Eck05], Definition 0.6). Let X be a compact Kihler manifold
with Kihler form w and let a be a pseudo-effective (1, 1)-class. A foliation F is called
numerically trivial with respect to a if for all 1 < p <; n - 1 and for all test forms
1
(X - SingF),
U E D(M-n-p
lim
sup
e40 TEa[--w]
f(Tac + Ew) /\ ul = 0
JX
where T varies over all currents with analytic singularities representing a with T >
-cw and where Tac is the absolutely continuous part of T in the Lebesgue decomposition.
Here, the condition T > -ew acts as a "perturbation" to ensure that T captures
the properties of nearby big divisors. Eckl shows that a admits a maximal numerically
trivial foliation F. The main result of [EckO5] is that the codimension of the leaves
of F is bounded by v(L), and furthermore if r(X, L) = v(L), then this foliation
coincides with the Iitaka fibration.
Similarly, there is a maximal fibration f : X -- + Z with fibers in F. Eckl calls
this the pseudo-effective reduction map and compares it to the notion of reduction
map in [BDPPO4].
Some of the arguments in [Eck05] are similar to the work in this chapter. In
particular, [Eck05] constructs a generic quotient in the context of foliations. All of
the arguments presented here were developed independently.
There are advantages to both the analytic and algebraic approach. [Eck05J is
more geometric, yielding insights into the numerical dimension for compact Kihler
manifolds. The algebraic approach yields a more explicit understanding of the pseudoeffective reduction map. It is also more suggestive of an approach to the problem in
characteristic p.
3.3
Generic Quotients
The pseudo-effective reduction map will be defined by taking a quotient of X by a
collection of families of curves. The geometry of the connections defined by these
families can be very subtle, as indicated by Example 3.2.3. In this section we will
show that the situation improves if we isolate the generic behavior of the families.
Suppose that X is a variety, and (Ui, Vi, si, wi) is a collection of surjective proper
families of subvarieties such that the fibers of si are generically irreducible. The
generic quotient can be described very succinctly: it is the map ir : X -- + Z found
by applying Theorem 3.1.5 to the open sets of the U on which wi is flat and si is
flat with irreducible fibers. In order to understand the properties of this map, we
will give a different description in Construction 3.3.5 using the quotient theorem for
proper families.
In order to study the generic quotient, we will need to understand how quotients
change under birational transformations. Suppose that # : Y -+ X is a birational
map. We first note that that points on X are connected through the families Ui iff
there are points above them on Y that are connected through the preimages of the
Ui .
Lemma 3.3.1. Let X be a normal variety, (Us, Vi, si, wi) a finite collection of proper
families of subvarieties. Suppose that # : Y --> X is a birational map. Then the quotient map associated to the U is birationally equivalent to the quotient map associated
to the families (U X X Y, Vi, si o P1, P2) on Y.
Proof. Let 7r : X -- + Z be the almost proper quotient map defined by the Ui, and
consider the composite ir o # : Y -- + Z. This is also almost proper, and it obviously
satisfies the criteria of Theorem 3.1.6.
F
In contrast, if we focus on the strict transforms of the subvarieties defined by
Ui, we expect the connections to change. Suppose (U, V, s, w) is a proper family
of subvarieties of X and that # : Y -+ X is a birational morphism. If we are to
have any hope of defining a strict transform family on Y, we should require that
w(U) is not contained in the center of any #-exceptional divisor. Assuming this,
we define the strict transform family f : U' -+ U to be the unique component of
U xx Y that dominates w(U). We have a natural map w' : U' -+ Y and projection
s
s o f : U' -+ V, and denote the new family by (U', V, s', w').
Example 3.3.2. Pick a point x E p2 . We denote the family of lines through x by
U. More explicitly, we define U := BlxP 2 , with s : U --+ PV the map with fibers the
lines through x, and w : U -+ p 2 the map exhibiting the fibers of the first projection
as the lines through x. The quotient map associated to U takes P2 to a point.
However, suppose we blow-up p 2 at x to obtain a new surface S. Then U Xp2 S
has two components; one is isomorphic to U and the other is a surface intersecting
U along the (-1)-curve representing tangent directions to x. The first component is
the strict transform family U', and the map w' : U' -+ S is an isomorphism. Thus,
the quotient by U' yields the natural map to P1.
The connections defined by strict transform families can be very subtle. However,
there is one case for which there will be no major changes.
Lemma 3.3.3. Suppose that X is a normal variety, (U, V, s, w) a surjective proper
family on X such that w : U -+ X is flat. Suppose # : Y -+ X is a birational
morphism. Then the the strict transform family U' on Y is equal to U xx Y.
Proof. Since flatness is preserved under base change, we have that P2 : U XX Y -+ Y
is a flat morphism. In particular, this means that P2 is flat (and hence open) when
restricted to each irreducible component of U xx Y. Since only one component of
U xx Y dominates Y, this implies that U xx Y is the same as the strict transform
E
family.
Lemma 3.3.1 then yields the following compatibility result.
Corollary 3.3.4. Suppose that X is a normal variety, (Us, V, si, w) a surjective
proper family on X such that wi : Ui -> X is flat. Suppose $ : Y -+ X is a birational
morphism. Then the quotient map defined by the strict transformfamily (U', V, s', w')
on Y is birationallyequivalent to the quotient map defined by (Ui, V%, si, wi).
In constructing the generic quotient, we will only deal with families for which
w : U - X is proper and surjective. This assumption means that we can define
the strict transform family for any birational map
#
: Y -+ X. More importantly, it
means that every family contributes to the generic quotient, so that the construction
has some nice properties. In view of the conditions of Theorem 3.2.5, we also require
that each si : U -> V have generically irreducible fibers. This requirement is justified
by Example 3.4.2.
Construction 3.3.5. Let X be a normal variety, (Ut, V4, si, wi) a finite collection of
surjective proper families of subvarieties such that si has generically irreducible fibers.
We construct a rational map r : X -- + Z which we call the generic quotient associated
to the Ui. Although we will make a number of choices during the construction, the
resulting rational map will be unique up to birational equivalence.
Lemma 3.3.3 guarantees that if the morphism wi : Ui -+ X is flat, then the
strict transform family of Ui coincides with the total transform. Thus, we may apply
Hironaka's flattening theorem (Theorem 3.1.4) to each of the morphisms wi : Ui -+ X
in turn to obtain a model V : X' -+ X such that every strict transform family
(UJ, Vi, s', w) has that w is flat. Let 7r' : X' -- + Z be the almost proper rational
map defined by these strict transform families. We define the generic quotient to be
the composition
7T =
7'
o
X
--
+ Z.
We first verify that the generic quotient is independent of the choices we made
during the construction. Suppose that a different sequence of flattenings yielded an
almost proper map 7 : X -- + Z. Corollary 3.3.4 shows that the quotients do not
change upon passing to a higher model. Thus, by passing to a common model of X'
and X we see the construction is unique up to birational equivalence.
We also check that the generic quotient captures the generic behavior of the connecting families. By resolving the map, we may assume that 7r' : X' -+ Z is a
morphism. Note that by Corollary 3.3.4, 7r' is still the quotient of X' by the strict
transform families Uj. Consider any open set Vo C Vi above which s' is flat with
irreducible fibers. Let U40 denote the preimage of V' under s'. Since w' is flat, w'uo
is open. We denote these open families of subvarieties by (U 0, 10, s+, w ) to distinguish it from the proper families. We can apply Theorem 3.1.5 to construct a rational
map r : X' -- + W. We show that this rational map is birationally equivalent to ir'.
Since the closure in X of a general fiber of T is an equivalence class defined by
fibers of the si, it is contained in an equivalence class defined by the s'. Thus -r
factors birationally through 7r'. Conversely, let F : X -+ W be a resolution of r and
let v- : U -+ X be a resolution of the family. Since the fibers of st are proper, the
strict transform of a fiber of si is contracted by-T o W. Thus, by the Rigidity Lemma
the same is true for every fiber in the family U -> V. This shows that r and 7T'
are birationally equivalent. As an important consequence, we see that two general
points of X' can be connected through the Uj' iff they can be connected using only
irreducible fibers.
Remark 3.3.6. Here we have defined a generic quotient for a finite set of families.
We can also define it for an infinite set of families using the following well-known
argument. Suppose we are given an infinite set {U,} eA of proper surjective families
of subvarieties. For a given finite subset I C A, we let r1 : X - -+ Z 1 denote the generic
quotient defined by the corresponding U. We then define the generic quotient of the
U. to be any quotient 7r1 such that Z1 has the smallest possible dimension. We need
to check that this is independent of the choice of subset I. Suppose that {U 1 } and
{Uj} are two finite subfamilies meeting the criteria. Consider the generic quotient
7~uj : X --
+
Z1Luj. Of course there are rational maps Z 1 -- + Z1uj and Z
--
+
Zpuj,
and by minimality of dimension these maps are generically finite. Since the quotient
is always a birational contraction, 71 and rj are birationally equivalent.
We also record the following consequence of Lemma 3.3.3:
Corollary 3.3.7. Let X be normal, (Uj, V,) si, wi) a collection of proper surjective
families. Suppose that # : Y -->X is birational. The generic quotient associated to Y
and the strict transform families Uj is birationally equivalent to the generic quotient
associated to X and the families U.
Finally, the following proposition will be useful later.
Proposition 3.3.8. Let X be normal, (Uj, VK, si, wi) a collection of proper surjective
families. Let 7r : X -- + Z denote the generic quotient. Suppose that $ : Y -+ X
is any birational map, and i' : Y -- + Z' the quotient obtained by applying Theorem
3.1.6 to the strict transform families. Then w' factors birationallythrough 7r.
Proof. Let (Uj, V4, s', w') denote the strict transform families on Y. As in Construction 3.3.5, let @ : X' -+ X be a birational map resolving the generic quotient. We may
assume by blowing up further that X' admits a morphism f : X' - Y. By Proposition 3.3.1, the proper quotient associated to the families (Uj X y X', 14, s$ oPi, P2,i)
is birationally equivalent to the quotient on Y. However, these families consist of the
strict transform families, along with some other components we have not tossed out.
Thus the corresponding map contracts at least as much as the generic quotient. 0
3.4
L-trivial Reduction Map
We now have one way of constructing a reduction map associated to a pseudo-effective
divisor L: take the generic quotient associated to all surjective families of curves C
with L - C = 0. In this section we will study the properties of this map. The main
result of the section is
Theorem 3.4.1. Let X be a smooth projective variety, and L a pseudo-effective line
bundle on X. Then there is some birational model # : Y -+ X, and an almost proper
map w : Y -- + Z with connected fibers satisfying
1. For a very generalpoint y C Y and curve C C Y through y with dim(w(C)) > 0,
we have $*L - C > 0.
2. For a general fiber F of 7r, we have v(L|F)
=
0-
X is a
3. The pair (Y, r) is the minimal quotient satisfying (1): if & Y
birational map and 2 : Y -- + Z an almost proper map with connected fibers
that satisfies (1), then - factors birationally through 7.
The pair (Y, 7r) is determined up to birational equivalence. We call it the L-trivial
reduction map. It only depends on the numerical class of L.
Proof (of Theorem 3.4.1): To construct 7r : Y -- + Z, take the generic quotient associated to all families of movable curves C with L - C = 0, in the sense of Remark
3.3.6. Clearly this map has Property (1). In Construction 3.3.5, we showed that
the generic fiber of ir is connected by irreducible members of our families. Thus by
Theorem 3.2.5 this map also has Property (2). Property (3) follows from Proposition
3.3.8, which says that the generic quotient is the minimal quotient as we look over
E
all strict transform families.
Example 3.4.2. It is necessary in Theorem 3.4.1 that the general member of our
families be irreducible. For example, consider a Hirzebruch surface Fe such that the
distinguished section CO has large negative self-intersection e. Let F be a fiber of
the projection Fe -I P1 , and consider the divisor F + Co. This divisor has Iitaka
dimension 1. However, it has vanishing intersection with the connecting family of
reducible curves (e - 1)F + Co.
Example 3.4.3. It is also necessary to work with dominant families. Consider again
Example 3.2.3, keeping the notation there. Let C denote the family of curves on X
defined by deforming C. Then C x xY has two components: the strict transform family
and an additional component lying above the exceptional divisor. By Proposition
3.3.1, the quotient of Y by both of these families gives the map to a point. Thus if
we allow non-movable curves in our quotient, we will run into the same difficulty as
before.
We begin by checking the basic properties of the L-trivial reduction map.
Proposition 3.4.4. Let X be smooth and L pseudo-effective. Let r : X -- + Z denote
the L-trivial reduction map.
1. If K(X, L) > 0 then the Iitaka fibration for L factors birationallythrough 7r.
2. If $ : X' --+ X is a birational map, then the #*L-trivial reduction map is
birationally equivalent to the L-trivial reduction map.
3. If L is nef, then the L-trivial reduction map is birationally equivalent to the nef
reduction map.
Proof. Let
#
: Y -*
X be a resolution of the L-trivial reduction map. Note that
0 < K(F, #*L IF) < v(F, #*LIF)= 0 for a very general fiber F of the L-trivial reduction
map. Thus the Iitaka fibration factors birationally through it, showing Property (1).
Property (2) is a consequence of the birational invariance of the generic quotient
as shown in Corollary 3.3.7.
To show Property (3), recall that the nef reduction map is constructed by quotienting out by all L-trivial curves. Thus, the nef reduction map factors birationally
through 7r. Conversely, recall that L is numerically trivial along a general fiber F of
the nef reduction map. Define a curve C by intersecting F with general very ample
divisors. The first property of Theorem 3.4.1 applied to C shows that the L-trivial
reduction map factors birationally through the nef reduction map.
LI
Proposition 3.4.5. Let X be smooth, L pseudo-effective. Let
7r
: X --
+
Z denote
the L-trivial reduction map. We have
K(X, L) < v(X, L) < dim(Z).
Proof. We just need to check the last inequality. Recall that V is invariant under
birational pull-back. Thus, we first make a birational transformation # : Y -+ X so
that the L-trivial reduction map 7r : Y -+ Z is a morphism. By Proposition 2.0.4,
if 7r : Y -+ Z is a fiber space and F a general fiber then v(L) < v(LIF)+ dim Z,
finishing the proof.
0
Example 3.4.6. Example 3.2.3 gives a good illustration of the L-trivial reduction
map. We will keep the notation established there. Recall that we have a smooth
threefold X, a line bundle L with i(X, L) = 1, and a connecting family of movable
curves C such that L - C = 0. In particular, the connections defined by the family of
curves do not reflect the geometry of L. However, if we pass from X to the birational
model Y then the strict transform family of the curves is no longer connecting. In
fact, two points are connected by the strict transform family iff they lie in a fiber of
the map g : Y -*
l.
Let us confirm that this is actually the L-trivial reduction map. Since g is the
litaka fibration for L, it factors birationally through the L-trivial reduction map.
Conversely, since two general points in a fiber of g can be connected by a deformation
of the strict transform of C, Property (3) of Theorem 3.4.1 shows that the L-trivial
reduction map is precisely g.
As discussed in the introduction, we would like to find the "maximal" fibration
such that v(LIF)= 0 for a general fiber F. A classical example due to Zariski shows
that the L-reduction map does not satisfy this property. We will see later that the
pseudo-effective reduction map corrects this deficiency.
Example 3.4.7. We recall Zariski's example of a surface S that carries a curve with
negative self-intersection which can not be contracted in the category of projective
varieties. Start by fixing a smooth cubic curve C in P2 . Since C is a cubic curve, we
can identify it with C/A for some lattice A. Furthermore, we can do this in such a way
that for any line H c P2 the sum of the points of HIc lie on the lattice. Pick 10 points
on C sufficiently general so that there is no non-trivial integral linear combination of
them that lies on the lattice. We denote their images in C by pi, . . . ,pio.
Let S denote the blow up of P2 along these 10 points, and let C' denote the strict
transform of C. Since C' has self intersection -1, we have that C' = N,(C'). We now
show that there is no irreducible curve T C S such that T - C' = 0. Suppose there
were such a curve. Clearly T can not be exceptional, so the pushforward fT must be
a divisor on P2. We set d to be the degree. Since fT and C can only intersect along
the 10 points chosen above, we see that dHIc ~ fTJc ~ E azp, for some integer
coefficients ai. This contradicts the generality of the pi.
Thus, the C'-trivial reduction map on S is the identity map. However, the map
from S to a point satisfies the defining Property (2) of the L-trivial map, showing
that the L-trivial reduction map is not maximal with respect to this property.
3.5
The Restricted Numerical Dimension for Curves
In order to improve upon the L-trivial reduction map, we must take the generic
quotient of a different set of curves. The key property of the L-trivial reduction map
is that v(LIF) = 0 for a general fiber F. Since we want to focus on this numerical
property on the fibers, the most natural restriction on our curves is vx 1c(L) = 0.
The main theorem in this section is the following:
Theorem 3.5.1. Let X be smooth and let L be a pseudo-effective divisor. Suppose
that C is a movable curve very general in its family. The following are equivalent:
1. VXic(L) = 0.
2. There is a birationalmap $ : Y -- X centered in B_(L) such that P,(#*L)-C' =
0 where C' denotes the strict transform of C.
Since numerical dimension is invariant under passing to P, it's clear that (2)
implies (1). For the other direction, we will need a lemma showing that we can
separate movable curves from the restricted base locus.
Lemma 3.5.2. Let X be smooth and L be a pseudo-effective divisor. Suppose that
C is a movable curve that is very general in its family. Then there is a birational
map @): Y -+ X, centered in B_(L), such that the strict transform C' of C does not
intersect B-(P,(#*L)).
Proof. Let w : C --+ X denote a generically finite family found by deforming C. Recall that by Theorem 2.2.3 B_ (L) has only has finitely many divisorial components.
The same is true for w- 1 (B_(L)): any divisorial component is either the preimage
of a divisorial component of B_ (L) or a w-exceptional divisor. Thus there is a birational map V : Y -> X centered in B_ (L) that extracts the image of any divisorial
component of w- 1 (B_(L)).
Let C denote the strict transform of C, and iii C -+ Y the strict transform
family. Since C is very general, the map C -> C is an isomorphisin on a neighborhood
of C. Furthermore, C only intersects divisorial components of w- 1 (B (L)). Thus C
is disjoint from B_(P,(*L)).
E
We can now finish the proof of Theorem 3.5.1.
Proof of Theorem 3.5.1: It only remains to prove the implication (1) =
that v 1 c(L) = 0 iff
inf P,(#*L) -0
0
(2). Recall
where # varies over all L-admissible birational maps and C denotes the strict transform of C.
Lemma 3.5.2 shows that there is a birational model # Y -> X centered in B_ (L)
such that C is disjoint from B-(P,(#*L)). Thus, the infimum described above is in
fact equal to P,(#*L) -C, proving the theorem.
D
3.6
Pseudo-effective Reduction Map
We will define the pseudo-effective reduction map for a pseudo-effective divisor L by
taking the generic quotient with respect to all families of movable curves whose very
general member satisfies vxlc(L) = 0.
Proof of Theorem 3.0.8: To construct 7r : Y -+ Z, simply take the generic quotient
associated to all families of movable curves such that vxic(L) = 0 for a very general
member C. This map obviously satisfies Property (2).
In order to prove Property (1), we will need to compare against the L-trivial
reduction map. First, identify a finite set of families of curves {CJ} that define the
generic quotient. By Theorem 3.5.1, for any particular family C, there is a model
: Xi -+ X such that P,(@7L)
.
C = 0 for the strict transform of a member of Ci.
Of course, this is also true for any higher model, so repeating the process we find one
model 0 : X -+ X so that P,($)*L) vanishes on every strict transform family. Then
the pseudo-effective reduction map is birationally equivalent to the P,($*L)-trivial
reduction map. This implies Property (1).
El
Corollary 3.6.1. Let X be smooth and L be pseudo-effective. There is some birational model # : X' -+ X such that the pseudo-effective reduction map for L is
birationally equivalent to the P,(#*L)-trivial reduction map.
Just as with the L-trivial reduction map, we can compare the pseudo-effective
reduction map against the Iitaka fibration:
Corollary 3.6.2. Let X be smooth and L be a divisor with rx(X, L) > 0. Then the
Iitaka fibration factors birationally through the pseudo-effective reduction map for L.
3.6.1
Properties of the Pseudo-effective Reduction Map
Proposition 3.6.3. Let X be smooth and L be pseudo-effective.
1. If $ : X' --+ X is a birational map, then the pseudo-effective reduction map for
L is birationally equivalent to the pseudo-effective reduction map for $*L.
2. If L is nef, then the pseudo-effective reduction map for L is birationally equivalent to the nef reduction map.
3. The pseudo-effective reduction map for L is birationallyequivalent to the pseudoeffective reduction map for P,(L).
In fact, by Corollary 3.6.1 these properties can be reduced to the corresponding
properties for the L-trivial reduction map.
Proof. To show (1), we apply Corollary 3.6.1 to see that both are birationally equivalent to the P,(f*L)-trivial reduction map for some birational model f : Y --+ X.
F
(2) and (3) follow from Corollary 3.6.1 and Proposition 3.4.4.
Y -+ X with
Example 3.6.4. Suppose that L has a Zariski decomposition $
$*L = P+N. Then the pseudo-effective reduction map for L is birationally equivalent
to the nef reduction map for P.
When the pseudo-effective reduction map for L is non-trivial there are important
consequences for the geometry of L.
Theorem 3.6.5 ([Nak04], V.2.26). Let f : X --+ Z be a morphism of smooth varieties
and L a pseudo-effective divisor on X such that i.(F,LIF) = v(F, LIF) = 0 for a
general fiber F of f. Then there exists a morphism g : W - T birationally equivalent
to f and a divisor D on T such that p*L ~( g*D+E where E < N,(p* L) is supported
on the components dominating T.
Thus, when r(X, L) 0 all the interesting geometry of L can be detected on the
base of the pseudo-effective reduction map.
We define the pseudo-effective dimension p(X, L) of L to be the dimension of the
image of the pseudo-effective reduction map for L. Note that p(X, L) is a birational
invariant of L.
Proposition 3.6.6. Let X be smooth, L pseudo-effective. We have
r,(X, L) < v (X, L) < p (X, L).
Proof. Since each quantity is invariant under birational maps, this follows from the
F
corresponding inequalities for the L-trivial reduction map in Proposition 3.4.5.
3.6.2
Abundant Divisors
Abundant nef divisors exhibit a number of nice geometric properties. In this section
we will show that abundant pseudo-effective divisors satisfy similar properties. The
main result is the following list of equivalent conditions for abundance. The equivalence of (1), (2), and (3) was shown in [Eck05], and the equivalence of (2), (3), and
(4) is proved in [Nak04]. Our contribution is to reprove the first set of equivalences
from an algebraic perspective.
Theorem 3.6.7. Let X be a smooth variety, L a divisor with n(L) > 0. The following
are equivalent:
1. ,(L) =v(L).
2. r(L) =p(L).
3. Let
f
:X' -- > Z' be the Jitakafibration for L with morphism p : X' -> X.
v(p*LIF) =
Then
0
for a very generalfiber F of f.
4. There is a birationalmap p : W -+ X and a contraction morphism g : W -+ T
such that P,(p*L) ~q g*B for some big divisor B on T.
We will first need a relative version of Theorem 2.1.25.
Definition 3.6.8. Let X be normal and f : X -+ Z a morphism. We say that a
movable curve C on X is f-vertical if f(C) is a point.
Theorem 3.6.9. Let X be normal, f : X -+ Z a morphism. Let k denote the
subcone C C NM 1 (X) consisting of curve classes that have vanishing intersection
with the pullback of an ample divisor H on Z. Then K is the closure of the cone
generated by classes of f-vertical movable curves.
Proof. Suppose the theorem fails. Fix an ample divisor A on X. There is a divisor
D (not pseudo-effective) such that
e D - a <0 for some a
CK.
* For some e > 0, D - C > cA - C for every vertical movable C.
If F is a very general fiber of f, then every curve C C F that is movable in F is also
movable in X. So Theorem 2.1.25 implies that the restriction of D to a very general
fiber is big. This implies that D is f-big, so that D + mf*H is pseudo-effective for
some large m. But (D + nf*H) . a < 0, a contradiction.
We now turn to the proof of Theorem 3.6.7.
Proof of Theorem 3.6.7:
(1) -> (2): All the quantities are invariant under birational maps. So we may
assume that the Iitaka fibration for L is actually a morphism f : X -+ Z. We
would like to show that v(F, L IF) = 0 for a general fiber F of f. By the maximality
r,(X, L), finishing the
of the pseudo-effective reduction map this implies p(X, L)
implication.
Recall that the numerical dimension v(L) is defined to be the minimal dimension
of a subvariety V such that s(L, V) = 0. Choose such a subvariety V. Note that V
dominates Z under f: if it did not, there would be some very ample H on Z such
that f*H contains V. But since f*H < mL for some m, we would have §(L, V) >
s(f*H, V) > 0, a contradiction. In particular, V n F is 0-dimensional for a general
fiber F of f.
Let # : Y -+ X be a birational map extracting V as a divisor (which we denote
by E), and let g denote the composition f o # : Y -+ Z. Since V dominates Z under
f, the restriction of E to a general fiber of g has codimension 1. Fix an ample divisor
A on Y. By definition there is some b such that xL - yE + A is not pseudo-effective
for any x, y > b. We will fix a specific y > b.
Choose any compact slice of NM 1 (Y), and let Q denote the closed region of this
compact slice on which -yE + A is non-positive. If L is positive on Q, then (by
compactness) xL - yE + A is positive on all of NM1 (Y) for x sufficiently large,
a contradiction. Thus, there must be at least one curve class a E NMI1 (X) with
(-yE + A) . a < 0 and L - a = 0. This means that E - a > 0. Also, since L > ef*H
for ample H on Z and sufficiently small e, we have f*H . a = 0. Thus, a lies on the
face of NM 1 (Y) cut out by g*H. By Lemma 3.6.9, we can find a sum of g-vertical
movable curves C := E aiCi that lies arbitrarily close to a. In particular we may
ensure E - C > 0.
Choose a general fiber G of g, and let F = #(G) denote the corresponding fiber
of f. By deforming the movable curves Ci, we may assume that they all lie in G.
Thus we may identify C as a curve class in NM 1 (G). Of course LIG - C = 0 and
EIG-a > 0, showing that g(L IF,V n F) = 0. Since V n F has dimension 0, this shows
that v(LIF) = 0, finishing the implication.
(2) => (3): Since all the quantities are birationally invariant, we may pass to higher
Z' and the pseudo-effective reduction
models so that the Iitaka fibration f : X'
map 7r : X' - Z are both morphisms. Since the Iitaka fibration factors birationally
through the pseudo-effective reduction map, after possibly going to yet higher models,
we find that there is a birational morphism g : Z -+ Z' such that f = g o 7r. But of
course a general fiber of f is also a general fiber of 7r, proving (3).
(3)
(4): This is a consequence of Theorem 3.6.5.
(4) 4 (1): It is clear that P,(p*L) is abundant. By Theorem 2.7.2, this is equiv0
alent to the abundance of L.
Condition (4) of Theorem 3.6.7 implies that abundant divisors satisfy some of the
same properties as big divisors. We will collect some of these here.
Continuity Results
Corollary 3.6.10. Let X be a smooth variety and L be an abundant divisor. Suppose
that D is Q-linearly equivalent to an effective divisor and satisfies D = L. Then D
is also abundant.
Proof. It suffices to show that t,(X, D) = ii(X, L). Recall that there is a birational
map p : X' -> X and an algebraic fiber space f : X' -> Z such that P,(pt*L) ~Q f*B
for some big divisor B on Z.
Note that P,(pt*D)|F - 0 for a general fiber F of f. Since D is Q-linearly equivalent to an effective divisor, P,(p*D)IF -Q 0. That is, r,(FP,(p*D)IF) = 0, SO
Theorem 3.6.5 implies that P,(p*D) ~Q f*D for some divisor D on Z. Since clearly
D = B, we obtain abundance of D.
E
Proposition 3.6.11. Let X be smooth, L abundant. Fix an ample divisor A. Then
any discrete valuation v satisfies
v(|IL||D = lim v
L+
A).
In particularthe asymptotic valuation of an abundant divisor is a numerical invariant.
Proof. The space of all discrete valuations is compact and the divisorial valuations
are dense in this space (see [VaqOO]). Thus we may assume without loss of generality
that v measures the order of vanishing along a subvariety T. We first analyze the case
when there is a flat morphism f : X -> Z with connected fibers such that L -Q f*B
for a big divisor B on Z. We set T' f-f(T). Suppose that Tj are the irreducible
components of T'. We define the function WTI : K(X) -> R to be WTI = min vr, so
that WTI measures the minimum order of vanishing along any component of T'.
Since the fibers of f are equidimensional, by intersecting general hyperplane sections we may find a smooth subvariety Z' mapping finitely onto Z under f. Since L
is pulled-back from Z, we have
vT(IILIJ) = WT/QILI)
=JT'nz'(jIL1z'/D
where WT'nz' measures the minimum order of vanishing along any component of
T' n z'. In general we have inequalities
vT(
Since
f
L + -A
)
w TA) L + -A
WTnz
( (L +
A)
is finite on Z', there is some H sufficiently ample on Z so that (f*H - A)Iz'
is ample. In all, we have
VT(IL
>i)WTI( LA-A)
(
(+
+-A
|WTnZL LA)
'M?
Z'
L + If*H
> WTZ'nz(
Thus it suffices to show that the last term converges to WTnz,( L Iz,)as m increases.
Let Z denote some component of T' n Z'. Since B is big, we can write B ~Q D + 5H
for some small 6. Then by the triangle inequality for valuations applied to (1+)L ~(Q
L+
f*H + If*D we have
vz|IL z||
I)
vz'
m1
(
L + f* H
+ m(
z, /
+ Ivz(f*Diz,).
m
Taking the limit as m goes to oc shows the desired equality for vz,, implying we also
have equality for WT'nz'.
We now turn to the general case. Since valuations can be computed on any
model, we may assume there is a flat morphism f : X -+ Z with connected fibers
such that P, (L) ~Q f*B for a big divisor B on Z. The earlier calculation shows that
v(l|P,(L)||) =limm- v(iiP,(L)+ All). Since P,(L + ±A) - P,(L) is big, it is also
true that
.
v(||P,(L)|| = lim v ( P, L+ -A
mj
m-oo
[Nak04] 111.1.8 shows that for a big divisor L we have v(IIBII) = v(1|P,(B)II) +
v(N,(B)). Using the triangle inequality, we see
v(IILII)
v(||P,(L)||) + v(N,(L))
P, (L+-A)
lim
"lim
v
+v
(N,
(L+-A)
L+-A
v(||L||
finishing the proof.
3.7
0
Relationship to the Minimal Model Program
The minimal model program gives a conjectural description of the litaka fibration for
the canonical bundle Kx in geometric terms. The main conjectures of the minimal
model program are the following:
Conjecture 3.7.1 (Existence of Minimal Models). Let (X, A) be a kit pair such that
Kx + A is pseudo-effective. Then there is some birational contraction # : X -- + X'
such that (X', #,A) is kit, Kx, + ,A is nef, and R(X', Kx' + #4A) = R(X, Kx + A).
Conjecture 3.7.2 (Abundance Conjecture). Let (X, A) be a klt pair such that Kx+
A is nef. Then Kx + A is semiample, i.e. there is a morphism f : X --+ Z so that
Kx + A ~R f*A for some ample divisor A on Z.
The essence of the Abundance Conjecture is that the asymptotic behavior of
sections of Kx + A should be predicted by its numerical behavior. This idea was
made precise in [Kaw85], which proves the Abundance Conjecture in the special case
when Kx + A is nef and abundant (see also [Fuj09]). The following slightly stronger
formulation occurs in [Nak04] and [BDPP04]:
Conjecture 3.7.3. Let (X, A) be a klt pair. Then Kx + A is abundant.
Note that the abundance of Kx +A would follow from Conjectures 3.7.1 and 3.7.2.
In fact, these two conjectures imply the following description of the Iitaka fibration.
Proposition 3.7.4. Let (X, A) be a kit pair, with X smooth and Kx + A pseudoeffective. Assume Conjectures 3.7.1 and 3.7.2. Then the Iitaka fibration for (Kx +
A), the (Kx + A)-trivial reduction map, and the pseudo-effective reduction map for
(Kx + A) are all birationally equivalent.
Proof. Since the conjectures imply that Kx + A is abundant, Theorem 3.6.7 shows
that the Iitaka fibration is birationally equivalent to the pseudo-effective reduction
map for Kx + A. Furthermore, since the map # : X -- + X' to the minimal model is
a birational contraction, #-- is an isomorphism on a neighborhood of a general curve
C contained in a general fiber of f. In particular, (Kx + A) - C = 0 for such a curve.
Thus the (Kx + A)-trivial reduction map also yields the Iitaka fibration.
E
Since the pseudo-effective reduction map satisfies nicer properties than the Iitaka
fibration, it can be used to study Conjecture 3.7.3. The key question is to understand
when the pseudo-effective reduction map is non-trivial. Thus, we make the following
conjecture:
Conjecture 3.7.5. Let (X, A) be a klt pair. Suppose that vxic(Kx + A) > 0 for
every movable curve C on X. Then Kx + A is big.
Our main application of the pseudo-effective reduction map is the following theorem:
Theorem 3.7.6. Conjecture 3.7.5 holds for all X of dimension at most n iff Conjecture 3.7.3 and Conjecture 3.7.1 hold up to dimension n.
This result is an extension of [Amb04], which proves a similar theorem in the case
when Kx + A is nef. The main point is that the results of [BCHM10] allow us to
contract components of N,(Kx + A), as was first noticed in [Dru09]. If we assume
that ,(Kx + A) > 0, we can use the recent work of [Lai09] instead. A similar result
appears in the recent preprint [Siu09].
Proof. Since the reverse implication follows from Proposition 3.7.4, it suffices to show
the forward implication. Let (X, A) be a klt pair. It is enough to consider the case
when Kx + A is pseudo-effective but not big. Thus by assumption there is a movable
curve on X with vxic(Kx + A) = 0.
Consider the (non-trivial) pseudo-effective reduction map for Kx + A. By Lemma
3.6.5, after passing to a new model # : X' -+ X we may assume the pseudo-effective
reduction map is a morphism f : X' -> Z and also that P,(4*(Kx + A)) = f*E for
some divisor E on Z. Now, we can write
Kxi + A'+(
aiEi
$*(Kx + A)+
bjFj
where the ai and bi are positive, A' is the strict transform of A, and the Ei and F
are #-exceptional and share no components in common. Since the Fj are exceptional,
we have
P,(#*(Kx + A) + (3 bF) = P,($*(Kx + A)).
(See for example [Nak04] 111.5.14.) After replacing (X, A) by (X', A' + E ajEj), it
suffices to show abundance under the additional assumption that there is a morphism
f : X -+ Z such that P,(Kx + A) is f-numerically trivial.
The next step is to use the arguments of [Dru09] to contract N,. Recall that
N,(Kx+A) has only finitely many components and each is contained in the restricted
base locus of Kx +A. Thus, there is some small ample A such that each component is
in the restricted base locus of Kx+A+A. By [BCHM1O], we can run the Kx+A+Aminimal model program over Z. As a result, we obtain a model V) X -- + X and
g : X -+ Z such that K 2 + A + A is g-nef. In particular, we must contract every
component of N,(Kx + A) that is not f-numerically trivial. Thus K 2 + A is gnumerically trivial.
Finally, we apply the arguments of [Amb04] to the resulting variety.
Theorem 3.7.7. Let (X, A) be a Q-factorial kit pair, and f : X -+Y a (Kx- + A)numerically trivial fibration. After passing to birational models f' X' -+ Y' (where
p denotes the birational map X' -+ X), we have
f'*(Pa (Ky, + A'))
=
Pa(p*(Kx + A))
for some A' on Y' such that (Y', A') is klt.
Note that abundance follows immediately by induction, proving Conjecture 3.7.3.
In particular, we have weak non-vanishing in dimension up to n, so the main result
E]
of [BirO9) implies Conjecture 3.7.1.
68
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